Misplaced Pages

Revision as of 19:49, 7 July 2007

	Integral was a good articles nominee, but did not meet the good article criteria at the time. There may be suggestions below for improving the article. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
Article milestones Date Process Result October 23, 2006 Good article nominee Not listed

Mathematics Start‑class Top‑priority

This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Start This article has been rated as Start-class on Misplaced Pages's content assessment scale. Top This article has been rated as Top-priority on the project's priority scale.

Template:WP1.0

To-do list for Integral: edit · history · watch · refresh · Updated 2007-11-09 Here are some tasks awaiting attention: Article requests : * the article seems to lack focus and order, and there is no table of contents. Also, brief discussions on the general properties of the integral such as being a linear functional, along with two brief sections on the two definitions of the integral. treatment of integrals with regard to differential forms. Some images to illustrate the Informal discussion section. Like what?--Cronholm 21:49, 28 June 2007 (UTC) A (sub)section on "Properties of integrals" covering general properties as a linear functional, Fundamental theorem of Calculus, etc. Copyedit : * Once major changes are complete, a thorough copyedit for flow and consistency is in order. Expand : * the section on Computing integrals could do with some expansion.

Archives

Index 1, 2, 3, 4, 5

Antiderivatives

Shouldn't we include a proof that integrals can be calculated through antidifferentialtion?

I'll see if I can write one —The preceding unsigned comment was added by Sav chris13 (talk • contribs) 05:10, 5 February 2007 (UTC).

Uh... I guess this was written before the fundamental theorem was in the article.--Cronholm144 12:06, 2 June 2007 (UTC)

Proof integral is anti-derivative

Let f(x) be a function

and F(x) be the function of the area under f(x)

Also note the relationships:

   Area = Length X Width

   Gradient = Rise / Run

The Rise in F(x) is F(x + h) - F(x) as h --> 0

The Run is h

The derivative of F being repressented by

   /h    as h --> 0

Now F(x) is the Area function

And values along the x-axis represent the "width" of our area

So h is the width of this area

So

   Gradient = Rise / Run = Area / Width = Length

Ie: /h = f(c)

For some value of c which is between (x + h) and x

Now as h --> 0 c will approach x

   /h = f(x)  as h --> 0

Hence the relationship between F and f is

   F'(x) = f(x) —The preceding unsigned comment was added by Sav chris13 (talk • contribs) 07:19, 5 February 2007 (UTC).

Are there any thoughts about this post? if not, I will archive it.--Cronholm144 12:08, 2 June 2007 (UTC)

You should say that this was discovered by Newton, and the others had some different proves (like Leibniz, and we should put their proves, I'll try to write Leibniz's, and Cauchy's proof). Some things you should change in your proof like Rise/Run, Area... You should put some graphic explanation. I'll do another version soon.

I'll log in later as CRORaf 195.29.73.31 11:03, 9 June 2007 (UTC)

If you wish to help with this month's collaboration, please make your edits (and comments) at the sandbox version. Thanks. --KSmrq 15:18, 9 June 2007 (UTC)

Horrible!

I linked to this article trying to explain what I meant by "integrating power with respect to time to get energy" but was horrified to find that within the first 12 lines of text we were already using set theory notation and Rieman definitions - before we even hit the table of contents. I strongly suspect someone who really understood the topic could explain it for the proverbial bright 12-year-old reader before disappearing into hihger maths; the basic concept deserves a more lucid explantion than this. I'm NOT a mathemetician but if someone doesn't come along and write a lucid introduction within the next few days, I *WILL* haul out my old maths books and write a better one. Painting a picket fence would be a good introduction to the topic - make the fence height variable, and make the pickets smaller and smaller...how much paint do you need? That sort of homey explantion before we get into the runes. --Wtshymanski 00:19, 9 March 2007 (UTC)

The introduction to an article is a place for summarisation, not the place to illustrate the topic with examples. That should be done in the first few sections of the article. An intuitive picture of integrals is given by the following sentence:- "The integral of a real-valued function f of one real variable x on the interval is equal to the signed area bounded by the lines x = a, x = b, the x-axis, and the curve defined by the graph of f." The basic content of your fence example is also presented in the nest line, but in beter language using graphs instead of physical objects.

It is not possible to say anything about calculus (or most other things in mathematics) without reference to set theory. It is a basic tool that any student must pick-up before he can hope to make further headway in mathematics. Area under the curve seems to be a pretty lucid explanation to me, more lucid than attempts to make the concept 'physical' or 'homey' by using fences and such. In any case, a student who wants to learn integral calculus from the basics will be better off using some textbook. We are an encyclopedia, not a textbook. We are here for reference, not for learning as such. In any case, in Science you should never try to oversimplify, few students need it. Loom91 06:39, 9 March 2007 (UTC)

I agree largely with Wtshymanski here. I can very easily see how the intro may not flow well for an unfamiliar reader. I agree with Loom91 that the intro is the place for summarization, but the summary should be clear to those unfamiliar with the topic. For instance, people throw the phrase "signed area" around as if this is a common concept. Give examples showing otherwise if you can, but I believe that the first time a student would see this phrase is in a calculus course, 'round about the time integrals are introduced (certainly in my teaching, that's the first time I would use the phrase). So, I don't believe that a sentence using this term is "intuitive". Improvements could be made by considering non-negative functions first, and including a figure. Similiarly, the phrase "measure of totality": what the heck is that? I know what it means, but would this help anyone unfamiliar with the integral idea? Finally, there is lots and lots that one can say about mathematics without reference to set theory: people (yes, even mathematicians) do it all the time. It may be a challenge, and may not be formal enough for some people, but I believe we need to strike a balance between formality and accessibility, especially in the intro. Cheers, Doctormatt 08:18, 9 March 2007 (UTC)

I agree about signed area being an unfamiliar concept, and it should be replaced by a brief explanation ("the integral is the area under the curve, with the caveat that if the curve drops under the x-axis then the area lying below must be subtracted from rather than added to the area lying above"). A figure is already included. And the phrase "measure of totality" was used after a previous discussion as something that gives an intuitive idea of integral at the most general level without oversimplifying. took it from Roger Penrose. We also have to consider our minimum target audience. Would it be fruitful to talk about integrals with a student who hasn't even picked up basic set theory yet? No set theoretic results are being used in the introduction, only basic set-builder notation that most students pick up within ninth grade. I learned calculus in ninth grade, and I see very few students wanting (and being able) to learn integrals before that. The conceptual jump from algebra to calculus requires a certain amount of mathematical maturity that only comes with age.

I also take a somewhat different view of the use of the introduction. Its main purpose is to inform a prospective reader whether he really wants to read this article. It must summarise the main features of the subject of the article, both to laymen and specialists. Explanations, the thing that non-specialists need, necessarily consumes space and is best left for the first few sections of the main body of the article, after which we can slip into technicalities for the expert who comes for reference. I think the latter is rather lacking in the article currently. Instead of relegating all material to the articles on the specific definitions, we should include brief discussions on the two most important definitions within this article. Some other things that are lacking are discussions on properties common to all integrals, such as its nature as a functional. Loom91 07:05, 12 March 2007 (UTC)

Latest edit: upright vs italic d

There is an anonymous user who is replacing italic d's with upright d's (for differentials, or the exterior derivative) in many articles. This is a point of view which I support. Both usages are common, the italic d being more common in the US, and the upright (roman) d being more common in the UK. As I am from the UK, my point of view may be biased (although in general I favour US spellings for math articles, especially fiber). However, I think the upright d works particularly well in wikipedia because of the unique mixture of math and wiki text in which it occurs. I therefore not only presume (as we all should) that this user is acting in good faith, but think this good faith is justified. Geometry guy 23:11, 22 March 2007 (UTC)

The anonymous user was me, and I must apologise for my misunderstanding and misconduct. I am new, and I hadn't yet contributed any constructive material to the encyclopaedia. I am very sorry to the editors who had to go around clearing up my unsolicited edits! I am yet to create an account, so until I do, my name is Simon. Since the community has been good enough to accept my intentions, I will simply put in my two pennies regarding why I did what I did, thank you for listening.

I am new to advanced mathematics, (and am therefore liable to be mistaken in some of my reasoning) but I find the upright d clearer for a number of reasons. I haven't been doing calculus for a huge amount of time but the way I have been taught I usually think of it as an algorithm for the manipulation of infinitesimals, and dx as a symbol suggests something slightly different going on (I find calculus very different!) rather than simply arithmetic. Of couse it also helps differentiate d * x. As a last thought, I think, in the context of the math formatting used on Misplaced Pages, it seems better in terms of aesthetics - this is, to me, quite important. Anyway, I'm sorry I didn't read up on policy before editing because I am certainly not the type of person inclined to force my methods on others. I plan to study engineering, but as yet I am rather new to more advanced mathematics. I am the first to admit that I am not familiar with the pros of using the italic d. I'm sure someone will enlighten me! Hope I have been more helpful than before.

Don't worry: there may have been misunderstanding, but certainly not misconduct. One great thing about wikipedia is that anything can be fixed using the edit histories (and it is dead easy to do, so no apologies are needed): hence wikipedians are encouraged to be bold in their edits! As for the italic d, it is a matter of convention (and is far from universal), going back to the way differentials were introduced, but conventions can change, and the young are the future! Geometry guy 20:18, 24 March 2007 (UTC)

Definition of Integral

I think we should consider including the definition of the integral in this article. That is,

the lim N->infinity of the summation from k=1 to N of f(a-kΔx)Δx, where Δx=(b-a)/N. This is describing the method of using an (approaching) infinite number of rectangles to produce the area of the function f in the interval .

This is in relation to integration on 2D planes. I am well aware that there are refined definitions for integrals of other circumstances, which should also be included. The reason I bring this up is because we include the definition of the derivative under its own section but only give the fundamental theorem of calculus under the Integration section. I feel this definition should be included.

If we could, I would like to discuss this and if others want, I could spearhead the initiative myself(of course, with the help of others).

Gagueci 20:39, 1 May 2007 (UTC)

Please do not post the same comment multiple times, it may be considered vandalism. I'm also in favour of including two brief sections on the two most popular definition of the integral here. Loom91 07:03, 2 May 2007 (UTC)

Sorry about the multiple posts, my message was not going through (so I thought) so I clicked save changes a few times. When I realized later that three had been posted, I deleted the other two. Gagueci 20:11, 3 May 2007 (UTC)

Broader coverage needed

As outlined in my rating comment, I think the scope of this article needs to be broadened to cover the concept of integral in appropriate generality, not concentrating only on integrals of real-valued functions of one real variable. While this is a critical special case (and indeed the key building block for other inegrals), it is by no means sufficient for an article that aims to cover one of the most critically important mathematics concepts.

However, care should be taken not to introduce better coverage at the expense of too high level of demands for readers — this is likely one of the most viewed maths articles. Therefore most technical details belong to either in later sections or in particular separate articles.

Additions and changes proposed include:

• Add multiple integrals or integrals of functions of several variables (a truly fundamental concept)
• Mention integrals of vector-valued functions (less elementary but important) and provide link
• Mention briefly integration of differential forms (and its somewhat more elementary version as countour integrals) and provide
• Better explanation of integral as a linear operator; its continuity properties
• More elaboration of integral as a (weighted) average; relation to expected value in probability / statistics
• Better references to Lebesgue integral and measure theory;
• Better pointers to applications;
• More focus in the list of various integrals: Lebesgue and one "simple" approach (Riemann or Daniell) + Stieltjes version should probably suffice for topics to be discussed in the article; pointers to other approaches could be made less prominently without bullet point lists;
• History of integral should be explained
• The lead should be compacted quite a bit and material moved to actual article sections.

Stca74 17:39, 15 May 2007 (UTC)

A very thoughtful and detailed suggestion. I agree completely on all points except the last one. I had proposed some of those before. I will further like to propose very brief discussions of the Rieman and Lebesgue integral in the article. Loom91 17:45, 15 May 2007 (UTC)

Agree as well. Integral is a Math colabortation of the week candidate. If it gets that it might get some more attention. --Salix alba (talk) 18:18, 15 May 2007 (UTC)

I also agree that this article needs to have broader coverage. I gave its companion article Derivative a similar treatment a month or two ago. This is what it looked like before: it covered only one real variable, lacked balance, and had a number of organisational problems, just like this article now. One practical suggestion I can make is to make better use of (and improve) subarticles: in a core topic such as this, one cannot cover all aspects in sufficient detail in one article.

I generally agree with the above suggestions, although I think it is particularly important to keep the perspective as elementary as possible and to provide an overview: specific topics (such as the list of various integrals) should be approached here from the viewpoint of the general reader, rather than the specialist in integration.

The down-rating to B-class is entirely appropriate, and possibly even generous: this is still a long way from being a good article. In particular, while Loom91 does not support a more compact lead, I am afraid there is zero probability of promotion to GA status with the lead as it is: see WP:LEAD. However, I have found that it is a wasted effort to try to write a good lead while the body of the article is unsatisfactory (for one thing, the lead should, to some extent, reflect the content of the article). So I suggest efforts should be focused on improving the main part of the article. The lead will then (again, in my experience) fall much more easily into place. Geometry guy 19:20, 15 May 2007 (UTC)

I think the article should remain simple. It used to have a comparison or Riemann and Lebesgue integration, and perhaps other stuff (I wrote a lot of that). Someone else took it out and over all I think that was a good move. It would be much better to explain the simplest concept of integration as well as possible and perhaps flesh out the links to the other integration articles. I think integration of differential forms does not belong in this article. Also note that it is probably futile to attempt to cover integration in full generality. The Itô integral, or integration with spectral measures for instance, does not belong in this article.

Loisel 05:00, 16 May 2007 (UTC)

I hadn't seen the reworked derivative article when I wrote my original comment here, but should I have seen it, I would have pointed it out as a model to follow — it has just the type of broad coverage at accessible level with generous links to other (sub-) articles that I had in mind. As for Loom91's comment, I partially disagree: the goal of explaining "the simplest concept of integration as well as possible", if done at the expense of broad coverage, is more appropriate for an elementary textbook (for Wikibooks?) than an encyclopaedia article. That said, I am also in favour of devoting more space for the more elementary concept, provided that the reader is made aware of the bigger picture. As for differential forms (and / or its more "elementary" versions), I still think they deserve a paragraph or two, with surely the bulk of exposition in a separate article. And the same applies to stochastic integral as well: it surely needs at least a one-sentence mention (how did I forget that?). Spectral measures I see rather as an application (of general vector-valued integration) than as a new integration concept as such, and thus would briefly mention under vectorial integration and point at spectral measure. Stca74 05:43, 16 May 2007 (UTC)

• Stca74, perhaps you are misattributing? I never said the quoted comment. I agree with you completely. Since this an topic overview in an encyclopedia (as opposed to a textbook), it should briefly mention the important parts of entire integration theory, including very advanced things like differential topology and stochastic calculus. If an article with the specific goal of making the entire content accessible to a layman audience (in this case, lower-grade students or people who did not have math as a subject in high school) is desired, that could be Introduction to integral. This article should be a general overview, with introductory content for laymen and technical content for experts. As for the lead, I can't find any disposable content in the current one. Every line seems essential. What could you remove from that? Loom91 10:56, 16 May 2007 (UTC)
  • Loom91, sorry, absolutely my mistake. The quote was from the comment above by Loisel, not from ´what you've written. Full agreement on the scope. As for the lead, I would second Geometry guy and leave the lead for now and review it once the actual article has been improved. Stca74 14:01, 16 May 2007 (UTC)

Where to put the dx

I've noticed that some people write ${\displaystyle \int f(x)dx}$ while others write ${\displaystyle \int dxf(x)}$ . Is there a story behind these two different conventions? --Itub 13:21, 21 May 2007 (UTC)

${\displaystyle dx}$ is an infinitesimal and can be treated as a normal variable; so both are the same and valid, but ${\displaystyle \int f(x)dx}$ is much more common. More of a debate comes over ${\displaystyle \int {dx \over f(x)}}$ v. ${\displaystyle \int {1 \over f(x)}dx}$. Dmbrown00 04:35, 31 May 2007 (UTC)

I don't think that both ways are good. ${\displaystyle \int f(x)dx}$ is good, but ${\displaystyle \int dxf(x)}$ means ${\displaystyle \int 1dx*f(x)}$ 195.29.69.197 09:35, 3 June 2007 (UTC)

I also dislike putting the differential first because, in Riemann-Stieltjes integration, it is unclear where the differential ends and the remainder of the integrand begins. JRSpriggs 10:06, 3 June 2007 (UTC)

When I was an undergraduate following a quantum mechanics course, a friend of mine asked me during the lessons: "Why does the Prof. write the d^3x before the integrand, what does this mean"  :).

Of course, this is just done for clarity. If you put the d^{n1}x1 d^{n2}x2...d^{nk}xk at the end, you are going to look at a function of many variables first only later to absorb the information which of the variables are integrated over.

Compare integration to summation. In a summation you write, say, "summation over k from zero to infinity". In an integration, if you write the dx at the end, you'll read it as, say, "integral from zero to infinity of blah blah blah blah blah....over x". Put dx first and you get: "integral from zero to infinity over x of blah blah blah blah blah....." Now, if the blah blah blah goes on for two pages, I think you'll prefer the latter notation :)

Also, if you write the integral as a repeated integral most of the dxk are going to move at their respective integral signs to the left anyway, unless you put delimeters around all the integrals to indicate which dxk belongs to which integral. Count Iblis 13:39, 3 June 2007 (UTC)

I tend to prefer the dx at the end because I feel it "closes" the integral, but it's true that for large or multiple integrals it can be a drawback. I've seen that putting the dx first is popular in some quantum mechanics books, so I was wondering if there was a story of how/when the two different notations became popular in different fields or communities. --Itub 09:11, 5 June 2007 (UTC)

Yes, we should try to find this out. It would be nice to include this sort of historic information in the wiki article. Count Iblis 13:13, 5 June 2007 (UTC)

Switching endpoints

Is this always true: ${\displaystyle \int _{a}^{b}f(x)\,dx=-\int _{b}^{a}f(x)\,dx}$? --Abdull 11:18, 24 May 2007 (UTC)

Yes. In general, (b-a) = -(a-b)Gagueci 19:07, 31 May 2007 (UTC)

For a Riemann integral, yes. For a Lebesgue integral, no (integral depends only on the measure of the set, it is not "oriented" in any way). — Preceding unsigned comment added by King Bee (talk • contribs)

2¢

In general a good article, but two points:

• Is that Arabic integral symbol a joke?
• Line integrals are never mentioned.

Dmbrown00 04:31, 31 May 2007 (UTC)

• no I don't believe so
• that needs to be corrected

--Cronholm144 12:12, 2 June 2007 (UTC)

Approximation of Definite Integrals section

This section must be elaborated on mathematically. Gagueci 19:11, 31 May 2007 (UTC)

More detail can be found in the numerical integration article. Oleg Alexandrov (talk) 01:28, 1 June 2007 (UTC)

Drone work

Since the first paragraph (indeed, the first sentence) frightens me, and the rest of the article needs massive work as well, I have concentrated my initial efforts on non-creative writing. Specifically, I have laboriously searched the Web and added some great references for the "History of integral notation" section. In doing so, I have established a precedent that I wish to be followed: Harvard style references with automatic links. I have yet to properly templatize (!) the Leibniz citation, but that is a minor issue, which I will fix Real Soon. (I plead fatigue.)

I'm not yet concerned with naming names within the bulk of the article, because I expect that to happen as it is pummeled into submission. I see a need for better coverage of the basics (linearity!), but we should also touch on contour integrals, complex integration, measure theory and analysis, differential forms, and (if we're really brave/foolhardy) de Rham cohomology or some such. --KSmrq 12:29, 1 June 2007 (UTC)

I second that. Specifically, I am sad to say that this article screams "I was written by a mathematician," I think a copy-edit by an English person would be a great boon. Also, I was giving the article a full read-through and I got to the part about integrals with more than one variable... then the article ended. What happened? Where is the rest of the article? I am surprised how an article that seems well written when given a cursory glance can lack so very much. I agree with KSmrq's assessment about the basics, so let's get working!--Cronholm144 11:35, 2 June 2007 (UTC)

P.S. I have created a sandbox User:Cronholm144/Integral for my more extreme edits. Anyone who wants to play is welcome.

The first paragraph is indeed appalling. It seems to have been written with an eye to generality, but devolves into verbose vagary to the point of being almost incomprehensible. I also note the full generality it seeks to cover is not actually dealt with in the article itself. In the meantime I've tried a rewrite of the first paragraph, which is the most glaring issue with the introduction. I would appreciate feedback, and hopefully we can make what I have even more accessible.

In calculus, the integral of a function extends the concept of a sum over a discrete range to sums over continuous ranges. The process of evaluating (or determining) an integral is known as integration. Integration is often used to find the "total amount" of a property on some bounded domain, whenever that property varies in a smooth or continuous fashion. For example: finding the temperature or electric field strength of a volume of space, since both temperature and field strength vary continuously in space; or finding the total acceleration or velocity over an interval of time, since both vary continuously in time. However, the mathematical treatment of integration is sufficiently general that it can be used to work with with any property that can be viewed as undergoing variation over a continuous domain.

I still feel it is a little vague, and I'm dicing with the issue of smooth vs. continuous (smooth, in a general sense, is more accessible to laymen, but it has specific mathematical connotations which may be worth avoiding here). Suggestions and feedback are welcome. -- Leland McInnes 00:06, 3 June 2007 (UTC)

I've dropped it into place in the article for now. I'll start trying to clean up the rest of the introduction soon. -- Leland McInnes 17:08, 3 June 2007 (UTC)

Should include footnote citation to proof regarding x^x

I noticed that there is a sentence in the article that mentions that it is possible to prove that ${\displaystyle x^{x}}$ has no elementary antiderivative. While I'm sure that's true, the way the sentence reads begs the question of how to go about proving it. So it seems like there really ought to be a footnote citation that leads interested readers to an actual proof. (It may even be included in one of the references at the end of the article, but if so there's no indication which part of which reference to go to for the proof.) Dugwiki 20:51, 1 June 2007 (UTC)

While I'm sure it's true that a reference would be nice, this is an incredibly blind and superficial observation. Please read the comment just before yours. The only references in the article at present are the ones I added in the history of notation section — and we are using Harvard references, not footnotes. You propose putting lipstick on a pig; what the article needs most is a lot of good writing, with sparse, appropriate references added as we go. That is, if anyone is interested. --KSmrq 01:19, 2 June 2007 (UTC)

Discrete vs. Continuous

Loom91 removed my introduction, citing the fact that a function does not need to be continuous to be integrable. I agree, but then that is not what my introduction said -- the claim is that the function should be defined on a continuous domain since a function defined on a discrete domain can simply be summed over a subset of the domain in the usual manner. Integration deals with the issue of extending this to continuous domains, and I feel this provides the most natural "intuitive" description of what integration achieves. I would hope that we could discuss the introduction instead of just reverting it -- if nothing else it is better than the existing introduction which makes almost no sense to anyone who isn't well schooled in what integration is already. -- Leland McInnes 21:07, 3 June 2007 (UTC)

Sandbox Skeleton

Hey all, I have created a sandbox skeleton for the new look of the article and Leland has kindly outlined his vision for the article on the talk page. Please direct all major edits that you wish to try before adding them into the article proper there. Hopefully the skeleton can be given flesh in the sandbox and then life in the article proper. Cheers--Cronholm 19:25, 4 June 2007 (UTC)

I'll add my voice here: the article will benefit from a broad overhaul, but there's no point leaving the page a mess while we do it. I've now put the outline in the sandbox skeleton, so please come and have a look -- there is plenty of material to be filled in; from history, such as early Greek efforts at integration, through to robust formal definitions of Riemann and Lebesgue integrals, and discussions of line and surface integrals, differential forms, and de Rahm cohomology. If all the collab of the month people can drop by the sandbox version and help fill out their area of expertise we can end up with a great article by the end of the month. -- Leland McInnes 19:35, 5 June 2007 (UTC)

Cleaning up the lede

I've tried to clean up the remainder of the lede, focusing on making things tighter. I feel things like examples should be deferred to the article itself (an informal discussion section at the beginning for example) where they can be properly fleshed out. An effort on that front can be found at the Integral (sandbox version) page; any help there would be appreciated. Still missing from the lede are:

• Discussion/mention of differing definitions of integral (Riemann/Lebesgue, etc.)
• Mention of fields of application.
• Mention of techniques for computing integrals.

As always, any help in adding that succinctly and elegantly to the lede is most welcome. -- Leland McInnes 16:14, 6 June 2007 (UTC)

Notice that "lede" is not a word. The word is "lead" as in Misplaced Pages:Lead section. JRSpriggs 10:01, 10 June 2007 (UTC)

It is actually: see News_writing. It is a traditional (archaic) spelling of "lead" originally used in journalism and printing to avoid confusion with the lead type of old printing presses. Quite a few Wikipedians use it. Like you, though, I think the conventional spelling is more appropriate in Misplaced Pages. Geometry guy 10:34, 10 June 2007 (UTC)

Informal discussion

I have added an informal discussion to try and provide some more accessible descriptions of integrals than the formal definitions. What currently stands is a first cut, and is perhaps a little long. It could also benefit from a few diagrams. Any and all assistance is welcome. -- Leland McInnes 19:54, 12 June 2007 (UTC)

There should definitely be an informal discussion part, but the example used is unfortunate:"The second difficulty, linked with the first, is there are not finitely many measurements of instantaneous speed". Measurements are in their nature always finite. We might make a lot of meauserements but it will always be a finite number. However, there is nothing wrong with the general idea. JKBlock 16:20, 18 June 2007 (UTC)

True, but I was thinking of the measurements as being "given" as by an oracle of some kind, rather than physically taken. The alternative is to suggest approximating a finite set of measurements by a function over a continuous domain, but that only complicates the issue. I might try and add soem clarification on this point though. -- Leland McInnes 14:50, 19 June 2007 (UTC)

Introduction

Why not introduce what the actual resut of an integral is in the introduction. Surely the whole point of an article is to sumarise, then explain. I dont see how just having what an actual integral looks like without its result is useful in the introduction.

If you are the anon who added the ${\displaystyle =F(b)-F(a)}$, and the sentence "F(a) signifies the integral at the upper limit, similarly for F(b)", then the issue is that this is "the result" of an integral only through the Fundamental Theorem of Calculus, and not, for example, the result according to any of the standard formal definitions (which would involve limits or supremums). Introducing this as "the integral" serves to introduce a degree of confusion as to what an integral actually is. Also, the sentence added doesn't appear to be correct as stated, according to the definition of integral (that is, the definite integral, as opposed to the anti-derivative) used in the article. I agree that some mention of the FtoC in the intro may not go astray, but I suspect it would be better couched in a summary of the history of integrals. -- Leland McInnes 15:08, 13 June 2007 (UTC)

Numerical quadrature

Mathematicians tend to concentrate on the calculus/analysis aspects, so I dodged the competition by rewriting the section on numerical approximation. References forthcoming. Kahaner, Moler, & Nash is one; Stoer & Bulirsch is another. Maybe also the new Dahlquist & Björck, which is currently available online. Suggestions welcome. Enjoy. --KSmrq 23:21, 14 June 2007 (UTC)

Simpson's rule would be a good addition to the picture, but the numerical approximation section is already pretty long, so it is up to you. Cheers.--Cronholm 23:50, 14 June 2007 (UTC)

As you noticed, one of the challenges was to say just enough, leaving a thorough study to the dedicated article. Simpson's rule adds nothing; technically, two Romberg steps are equivalent. If I had to reduce the section to one thought, it is that the calculus-book idea of a rectangle rule is practically worthless; serious modern algorithms typically use some form of adaptive Gaussian quadrature, which is vastly more accurate and efficient. Maybe eventually I'll find a way to make this vital point more briefly and more clearly, but the pre-rewrite version did not make the point at all! --KSmrq 02:29, 15 June 2007 (UTC)

Lebesgue vs Riemann image is correct

Hi KSmrq, I understand why you would have doubt about this image because it's not clear that using simple functions corresponds to using horizontal slices, but that is really what's going on. Loisel 02:45, 17 June 2007 (UTC)

Sorry, no; it's not helpful. I refer you, for example, to this discussion (already alluded to in my edit summary). Please do not restore the image again without discussion. To that end, I'll ask WT:WPM for a broader spread of views, in case you and I alone are too close to see clearly. Thanks in advance for your patience. --KSmrq 17:23, 17 June 2007 (UTC)

Well that is certainly how I teach the Lebesgue integral in the real analysis class. So you've got a contingent of mathematicians who think that this is the correct picture. Loisel 18:41, 17 June 2007 (UTC)

Though I've no knowledge of Lebesgue integrals, I agree with Loisel that horizontal slices is how most textbooks represent the Lebesgue integral. Loom91 08:40, 18 June 2007 (UTC)

Really? Which book? I have 3 Real Analysis books, and I have never seen the horizontal slice picture in any of them. –King Bee  10:29, 18 June 2007 (UTC)

Many books wouldn't have a "discussion" because of the level of the material, but one of the standard books is Folland's Real Analysis, and it does have a discussion. To quote from my edition, page 56: "In particular, if one picks the sequence constructed in the proof of Theorem (2.10a), one is in effect partitioning the range of f into subintervals I_j..." Perhaps we can add this quotation and a note (see Fig. (XXX)) and the figure that was deleted. Loisel 21:02, 18 June 2007 (UTC)

Ask if you need help adding references; we're using Harvard, not footnotes. (The numerous instances already in the article demonstrate the necessary incantations.) --KSmrq 22:16, 18 June 2007 (UTC)

(unindent) No, KSmrq, you are using Harvard rather than footnotes. The article can be developed this way if you wish, but you must not revert other editors contributions because they do not conform to this style. If you wish to impose this style on the article (on the acceptable grounds that you were the initiating editor), you should edit contributions which do not conform, not delete them. This is simply courtesy, and I believe you are a courteous editor. Geometry guy 23:03, 18 June 2007 (UTC)

Are you trying to pick a fight?! We are using Harvard style, and I intend to be a pest about it. I'm perfectly happy to help Loisel, as I indicated in both my edit summary and here, should I be asked. At the moment the article has a list of 11 citations listed under "References", and I spent quite some time researching and formatting them, so the burden of courtesy lies on those stragglers adding a new reference.

I notice that the "footnote" in History was introduced when text was imported from the sandbox version. That was a mistake, of course; and the inattention to form is accompanied by an unacceptably sub-par citation. (Have you looked at the web page cited?) So I just deleted the whole thing. If someone wants to add substantial history references, I'd be delighted. I might even offer to help!

Meanwhile, I noticed you were kind enough to do some of the formatting for Loisel; so I went to the trouble to finish what the two of you started. I found and added the publisher, the edition, and the ISBN for Folland. (There is a second edition dated April 7, 1999 (ISBN 978-0-471-31716-6), but since the contents are not viewable online I did not feel at liberty to alter the citation.) And I moved the page number to the referencing context, which is where this one belongs. Since I am a courteous editor. --KSmrq 02:01, 19 June 2007 (UTC)

You are both courteous and skilled editors... there seems to have been somewhat of a misunderstanding of intent, but we are all working towards the same goal, Right? --Cronholm 02:28, 19 June 2007 (UTC)

Indeed, and thanks! This was just a friendly chastisement, because I know KSmrq has high standards, and indeed his response was admirable. It was a pity I missed the chance to join in the drinks and fine sentiments: mine would have been a long cold beer ;) Geometry guy 11:11, 19 June 2007 (UTC)

Improper integral

The statement beneath the improper integral: "An "improper" integral; when x = a is a point where f(x) becomes infinite." is argueable. f(x) does not 'become infinity'; the improper integral (in this version) is used when the function is not defined at x = a, and that's "why" the limit is used.
Also I think the notation
${\displaystyle \int _{a}^{b}f(x)dx=\lim _{c\to a}\int _{c}^{b}f(x)dx}$
(with a note that a is either a real number or infinity) is better since it covers both integrals where the function isn't defined at a point and integrals on unbounded intervals. Please comment. I'm looking forward to joining the mathematics project on wikipedia :) JKBlock 20:00, 17 June 2007 (UTC)

A very good suggestion. Feel free to implement it. Loom91 08:39, 18 June 2007 (UTC)

I've changed some of the others things on this subject as well (for instance it also said that for proper integrals the function had to be continuous (which is simply wrong)). Could someone else have a look at what happens when integrand is infinite, I don't know much about that case? What do you think of "The improper integral often occur when the range of the function to be integrated is infinite."? I think it should be left out as IMO it's used more for unbounded intervals. JKBlock 14:03, 18 June 2007 (UTC)

• I added a picture I created that depicts an integral with an infinite integrand (the integral is not convergent). I think it adds to the article to have both types depicted, but you can remove it or ask me to change the picture in some reasonable way, if you desire. Just happy to be on board! –King Bee  04:49, 20 June 2007 (UTC)

Darboux integral

Don't you think it would be a good idea to link to the Darboux integral at the definition of the Riemann integral? They are equivalent and the Darboux definition besides from being more 'strict' is relatively easy to use in proving some of the base properties of the integral (just don't try to integrate most functions directly with the definition :). I'm sorry if I "talk" to much but I need a little experience in when to edit. JKBlock 20:19, 17 June 2007 (UTC)

Removed a Harvard reference to Rudin's book

I removed a reference to Rudin's book in the beginning of the formal definition of Lebesgue integral. It was backing up a statement that the theory of Lebesgue integration is grounded on measure theory. However, this statement is hardly contested, and in having regard to the general level of foot notes / Harvard references in the article does not need a backup. Instead, the typography of Harvard references made the sentence look (to a hypothetical readed not knowing the history of the subject) like Lebesgue integral was only made possible by measure theory developed by Rudin in his book. Hence removed the reference. Stca74 20:39, 19 June 2007 (UTC)

Thanks for the careful read. My intent was not, however, what you infer; I simply chose an apparently awkward place to insert some kind of reference for the whole section. (Indeed, the relevance of measure theory is uncontested!) So I have restored the reference, but hopefully in a happier home. Feel free to push it around to where you think it best serves the intended purpose. --KSmrq 00:32, 20 June 2007 (UTC)

two remarks

I collaborate by (gently) criticising two things, hope that's OK:

• Firstly, I don't understand the phrase "A general k-form is then a vector space with basic k-forms as the basis vectors," (in the section on integration of differential forms). Rather I'd say a k-form is an element of the vector space spanned by basic k-forms and functions as scalars?
• Secondly, the visualization of the several approximation methods is nice, but especially the Romberg method is impossible to get an idea of by looking at the image. It's just too small. (The main article on the method gives the formulae, but not the link to the image). I would suggest to make four separate images out of the existing one, a little bit bigger and the text should refer to it (which is the case in the other 3 methods, but not that much in the Romberg one) Jakob.scholbach 15:39, 24 June 2007 (UTC)

Another thing: at the moment the advantage of the Lebesgue integral over the Riemann integral does not become clear enough. I miss a statement like: "every Lebesgue-integrable function is Riemann-integrable, and then the two integrals agree. However, there are Lebesgue integrable functions (e.g. characteristic function of the rationals) which are not Riemann integrable. Also, I assume that there are some less esoteric advantages of the extra-generality given by the choice of a measure(?). If so, this might be good to add. Jakob.scholbach 16:44, 24 June 2007 (UTC)

A reference for the fact that x (or similar examples) has no elementary antiderivative would be good -- a reference for this fact seems to me as least as important (on an English WP article) as giving a reference for the arab integral sign. Jakob.scholbach 17:42, 24 June 2007 (UTC)

About your Riemann/Lebesgue comment: It should be noted that all properly Riemann integrable functions are Lebesgue integrable, but there do exist improperly Riemann integrable functions that are not Lebesgue integrable. For this reason, great care should be taken when saying things like "all Riemann integrable functions are Lebesgue integrable," just so that no one gets confused. –King Bee  18:33, 24 June 2007 (UTC)

(editconflict)

X falls under Misplaced Pages:Scientific_citation_guidelines#Uncontroversial_knowledge. I am not sure what you mean by less esoteric, Lebesgue integration only comes up in 300 and 400 level(and beyond) math courses in college, so it might be a touch difficult to bring up a simple example as you suggest, although I believe Chan-Ho had an idea in this regard.

Ksmrq, will probably address your concern about the graphs. I don't quite understand your "and functions as scalars" addition, it is the 0-forms that act as the field of scalars, but I think this ought to be addressed by another editor. Cheers and thanks for your comments--Cronholm 18:38, 24 June 2007 (UTC)

@King Bee: OK, I didn't know this. Then, of course the statement has to be made more carefully. But still, it ought to be there in some form.

@Cronholm: Well, in a way lots of things which are in the article seem to be uncontroversial knowledge. I don't disbelieve the mentioned statement. Wishing a reference was more in order to have a "further reading" where one could learn more about the fact/generalizations/explanations etc. The other thing: I wanted to say something like: a k-form is an element of the K-vector space generated by basic k-forms, where K is the field of smooth (or C^\infty) functions. Finally, by less esoteric I meant an "honest" function (not something like char. function of the rationals) possibly in conjunction with some non-standard measure, bref, something with an application outside maths, perhaps in physics. Jakob.scholbach 19:03, 24 June 2007 (UTC)

A practical application of the sort you seek may perhaps be found in probability theory, where Riemann integration often does not suffice. Loom91 19:50, 24 June 2007 (UTC)

Hmm... I don't know if we have have an article about functions that do not have elementary antiderivatives, we probably should, and this way the citation worry would go away. As for the honest function, not that I am aware of. Someone probably is aware of one though, and they will likely comment here soon. --Cronholm 19:34, 24 June 2007 (UTC)

Jakob, thanks very much for taking the time to read and criticize the article!

1. With regard to k-forms, you are quite right that an individual form is not a vector space! Is that the essence of your complaint?
2. With regard to Romberg integration and the image, bear in mind that we use thumbnails like this in kindness to our readers with low bandwidth and/or small screens. It links to a much larger image that should have adequate detail. The text devotes a paragraph to each of the four methods illustrated. However, I confess I found it difficult to illustrate the interpolation used by Romberg in the same style as the other three methods; the choice I made really doesn't adequately convey the role of extrapolation. Perhaps I can try again after I'm finished with some other images. The vital idea of the section, both text and illustrations, is the importance of using a numerically sophisticated method like Gaussian quadrature rather than the rectangle method found in a basic caculus text. Did that come through?
3. I share your concerns about the content of the Legesgue discussion; it should clearly state why we seek a more sophisticated integral and what it buys us. I haven't had a chance to do anything about it yet, and since others have been looking at it I've been hoping someone else would beat me to it! Volunteers?
4. With regard to x, the article could use more helpful references (and links?) overall. As well, you'd be surprised how tough it can be to come up with good examples sometimes. For instance, I'd like to illustrate improper integrals of both types (infinite range, infinite domain) with one function, and my best effort so far is x, integrated from zero to infinity.

Note that an improper Riemann integral with an infinite domain can handle some oscillatory functions. For technical reasons, the Lebesgue integral does not exist if the absolute value does not converge, which is more restrictive. The Henstock-Kurzweil integral can handle any real function that either of these two can swallow, but it does not generalize to other domains as nicely as Lebesgue measure. So why is this not in the article? Maybe soon it will be; these things take experts and effort!

Again, thanks for your attention. --KSmrq 22:02, 24 June 2007 (UTC)

1. Yes.

2. OK, the thumbnail is one thing. If I would create an illustration, I would strive for one making a clear "statement" which comes through even without a magnifier. Here, I guess, it is not even really the size which is a problem, more the 3D-ish layout of the Romberg figure, which, in this size is absolutely impossible to see, and even in the magnified view it does not become that clear, I'm afraid. Why not separate the 3 layers into 3 images horizontally side by side. I could also imagine a animated gif, if it is possible that the animation starts only when the user clicks on the image. Yes, the difference between the methods comes out, but it remains a bit obscure why the Gaussian method is actually better. "a bit of luck" is not fully satisfactory to me, but perhaps this should then be covered at Gaussian quadrature. Another suggestion: remove the list of function values and replace it by a bigger graph, where the axes are labelled. Right now, the list contributes very little to the text, only the fact that the integrals are taken from -2.25 to 1.75 is used in the text. Also, names of variables which do not occur later, like the h in rectangle method, and the T(h_k) in Romberg, should be omitted, as they are rather confusing. Finally, I think one should not take an example where certain methods work very well (or even exactly) by chance (or luck). A more detailed discussion on the subpages would probably be the good place for a double-example like the one here (first the function in the "good" range, then the "bad" range). Here, one "generic" function should be enough to show the general advantages and disadvantages of the several methods.

4: Yes, WP is good in bringing all of us back to down-to-earth mathematics! e^(-abs(x)) is a little bit more compact than your function, gives improper integral 2 (-infty til infty).

Jakob.scholbach 01:34, 25 June 2007 (UTC)

Ahh, but ${\displaystyle e^{-|x|}}$ only gets as big as the value 1, and is bound below by 0; I believe KSmrq above was itchin' for an integral that is improper in both senses (in domain and in range). =) –King Bee  02:09, 25 June 2007 (UTC)

Correct. It is trivial to do one or the other; x is the standard example, for a fixed p. If p > 1, the integral from 1 to ∞ converges but the integral from 0 to 1 does not; if p < 1, the integral from 0 to 1 converges but the integral from 1 to ∞ does not.

Jakob, I'll try to revisit the Numerical quadrature section when I'm done with the rest. I share some of your concerns, but I had already spent a great deal of time on that one section and felt it was more important to move on to others.

The "accident" example is no accident; it is there to demonstrate the practical importance of insight into specific tasks. Another such example would be a periodic function, which is quite common in applications; but I felt that would be too complicated to present. In fact, I spent a long time looking for (a piece of) a function with a simple formula, a simple integral, a picture with certain properties I wanted, and so on. My intent is to use this as a running example throughout the article (except for improper integration). Consider that the current lead picture does not include negative regions, nor any use of color.

Thanks again for more thoughtful comments; I'll try to do them justice, starting with fixing the "forms" problem. --KSmrq 02:44, 25 June 2007 (UTC)

Some additional comments: I found a nice example of a function Lebesgue can't handle. As I mentioned before, Lebesgue has problems with oscillations. Let

${\displaystyle F(x)={\begin{cases}x^{2}\cos {\tfrac {\pi }{x^{2}}},&0<x\leq 1\\0,&x=0\end{cases}}}$

Then the derivative of F exists and is finite in , but is not Lebesgue-integrable in . For, let

${\displaystyle {\begin{aligned}a_{n}&{}={\sqrt {\frac {2}{4n+1}}}\\b_{n}&{}={\sqrt {\frac {1}{2n}}}\end{aligned}}}$

so that

${\displaystyle \int _{a_{n}}^{b_{n}}F'(x)\,dx={\frac {1}{2n}}.\,\!}$

The intervals are pairwise disjoint; thus if E is their union, then

${\displaystyle \int _{E}F'(x)\,dx\geq \sum _{n=1}^{\infty }{\frac {1}{2n}}=\infty .}$

This example is given in Behnke, Bachmann, Fladt, & Süss (eds.), Fundamentals of Mathematics, Volume III: Analysis (ISBN 978-0-262-52095-9), pp. 462–463.

To challenge Riemann integration, calculate the balance point (horizontal center of mass) of a steel beam with a ball bearing resting somewhere on top. The problem, of course, is the need for a Dirac delta function to accomodate the point contact of the ball.

I also forgot to mention that the table used in Numerical quadrature is carefully formatted to illustrate the way power-of-two partitions with the trapezoid rule in Romberg integration can recycle evaluations, something Gauss points cannot do (though Kronrod points help). And may I just add, getting that formatting to work was a huge pain! --KSmrq 22:02, 25 June 2007 (UTC)

Thanks for taking all this pain... It seems to be an ubiquitous experience, Pein in German, peine in French.Jakob.scholbach 23:25, 25 June 2007 (UTC)

For the sake of completeness, I suppose you should mention that while the Lebesgue integral of ${\displaystyle F'}$ (the function discussed above) does not exist, the Riemann integral does; it is equal to -1. –King Bee  12:48, 26 June 2007 (UTC)

In response to Jakob.scholbach's critique of the Romberg integration piece of the numerical quadrature figure, I have incorporated an exotic new element. It makes the image even busier than before, but it explicitly depicts use of the Richardson extrapolation polynomial. I'm very proud of myself for my graphical ingenuity, and for being able to explain the image in terms SVG can handle. My exuberance is tempered by a concern that the image may speak clearly to the already-enlightened, but only confuse those on the path to enlightenment. The typical numerical analysis text presents only formulas and tables of numbers, in part (I suspect) because that is the "coin of the realm", and in part because such images don't come easy.

Keep in mind that Romberg integration combines several layers of complexity. At the bottom is use of the (composite) trapezoid rule, and a formula for its error. Layered on that is exploitation of power-of-two partition refinement, which has two advantages: (1) old function values can be reused, and (2) interpolation is simplified. The third layer is Richardson extrapolation, a "deferred approach to the limit", which interpolates a Lagrange polynomial through the (h,T(h)) pairs and extrapolates to h = 0.

When I look at the image, I see all three components. Do others?

In any event, the end of the month has come, and I have not spent nearly enough time organizing and improving the many sections of text. I shall continue a little longer in that effort, and perhaps insert one or two additional images I've been working on. --KSmrq 09:36, 2 July 2007 (UTC)

Are you working on a GIF file? The animation was rather good for pedagogy.--Cronholm 12:46, 2 July 2007 (UTC)

I take it you're referring to a different image, the Riemann sum, yes? Already I have replaced the GIF with one static image, which more clearly depicts a tagged partition and the sum it generates. However, I would like to have as well an image showing increasingly finer meshes. For me, GIF animation is not an ideal solution, for several reasons: 1) no start/stop/rewind control, 2) limited color depth, 3) no resolution scaling, and in this case 4) no simultaneous view. The GIF I displaced has problems beyond that: a) it is gray, b) yet the gray areas are not solid but dithered, c) the microscopic rapidly changing number reads poorly, d) it makes comparison of fineness impossible, and e) it does not use the running example integral. Many calculus books, such as Stewart's (ISBN 978-0-534-39339-7, see also website), have illustrations that do a better job without animation. So my concept is a static image showing a series of finer divisions and the approach to the limiting value. (As SVG implementations begin to support animation better, I may begin to post SVG efforts.) --KSmrq 22:06, 2 July 2007 (UTC)

A convergent improper integral which is improper in "both senses"

Since KSmrq above was looking for such an integral, I have one for him (and anyone else ambitious enough to evaluate it):

${\displaystyle \int _{0}^{\infty }{\frac {x^{-a}}{x+1}}\ dx,\ 0<a<1}$

Perhaps you remember this beast from studying complex analysis. (You must integrate along a branch cut in order to work it out). Is this what you were in the mood for, or would you rather have something that can be evaluated using more elementary techniques? –King Bee  19:16, 25 June 2007 (UTC)

Thanks! This will do fine. A plot of ⁄_(x+1)√x looks good, and the integral from 0 to ∞ is exactly π. --KSmrq 21:31, 25 June 2007 (UTC)

Glad to help out. –King Bee  21:49, 25 June 2007 (UTC)

• Quick question - were you planning to make a graphic of the integral in question and insert it into the page? If so, where? If you don't feel like making a graphic, I can make one (similar to the other one I added to the page recently). –King Bee  12:44, 26 June 2007 (UTC)

  Quick answer: Done. --KSmrq 22:10, 26 June 2007 (UTC)

Having updated the figure for the improper integral section, I then made the mistake (?) of considering the text. I've tried to say a bit more without saying too much. One motivation is to explain the figure; another is to have enough to work with when comparing, say, Riemann and Lebesgue. I came perilously close to Cauchy principal values (within an epsilon radius?), but chose to say nothing at this point. Where are all the good editors when you need them? ;-) --KSmrq 07:17, 27 June 2007 (UTC)

To be perfectly honest, I have no idea how you are computing that integral. Where is the \arctan(x) coming from exactly? –King Bee  11:14, 27 June 2007 (UTC)

The derivative of arctan(u) where u is equal to x. He skipped a bunch of steps.--Cronholm 11:39, 27 June 2007 (UTC)

Ahh. Well, I thought that the only way to do these kinds of integrals was by contour integration, but I guess this one works out rather nicely just using the second fundamental theorem; this is why I was confused. –King Bee  11:43, 27 June 2007 (UTC)

You mentioned the general case of an exponent 0 < a < 1, which is usually nastier; but the a = ⁄₂ case is special and pretty. It is also sufficient for our purposes, and no one needs to see how we found an antiderivative; yet anyone who wishes can check that the derivative of 2 arctan(√x) is indeed 1/((x+1)√x). With this in hand, we can write closed form expressions to take to the limits, which is a Good Thing for our readers. Of course, I'm hoping that the explicit examples add to, rather than detract from, their understanding. --KSmrq 18:16, 27 June 2007 (UTC)

Oh yeah, I'm all about your example. It was just early in the morning and I glanced at the Integral page and found myself completely dumbfounded by your example, since I had just worked out that integral for general a between 0 and 1 the day before. I thought you had done something amazing and beautiful, and that I could forgo contour integration; however, your choice of a = 1/2 is indeed sufficient and pretty enough for our purposes. All's well on my end. =) –King Bee  18:52, 27 June 2007 (UTC)

I thought I had done something amazing and beautiful — not solving the general integral, but massaging the section. ;-) For the record, Mathematica gives a closed-form antiderivative with a free in (0,1) as

${\displaystyle -{\frac {x^{1-a}\,_{2}F_{1}(1-a,1;2-a;-x)}{a-1}},}$

where ₂F₁ denotes a hypergeometric function. Imagine throwing that at the kiddies!

Actually, just for cultural orientation maybe the article should give an example somewhere of how a modest little integrand can produce a scary sprawling result. Calculus book examples are seriously misleading in lulling students into thinking most integrals have tidy little closed form results; tables of integrals are not much better. --KSmrq 22:29, 27 June 2007 (UTC)

History of integration

Take a look at Henri Lebesgue#Lebesgue's theory of integration. It is not much good as biographical information, and not a great history either, but it prompted me to wonder whether we need a History of integration article. Neither Integral nor History of calculus contain any substantial information post 1850 (say), whereas a lot happened since then! Any thoughts? Geometry guy 18:04, 27 June 2007 (UTC)

The history section of this article might be augmented and polished. An article devoted to the history of integration might be interesting reading; it will surely be time-consuming to write. Let's do what we can here, and if it gets too long spill over. --KSmrq 08:20, 28 June 2007 (UTC)

Example of nasty integral

In the interest of giving first-year calculus students a healthy jolt of reality, I'm looking for a really simply integrand with a really complicated antiderivative. A mild example:

${\displaystyle \int {\frac {dx}{x^{5}+3x^{3}+8x^{2}+24}}={\frac {-14{\sqrt {3}}\tan ^{-1}\left({\frac {x-1}{\sqrt {3}}}\right)+64{\sqrt {3}}\tan ^{-1}\left({\frac {x}{\sqrt {3}}}\right)+26\log(x+2)+36\log \left(x^{2}+3\right)-49\log \left(x^{2}-2x+4\right)}{2184}}}$

I'd like to stay away from special functions in the result, but still I'm sure we can find nastier examples. Suggestions? --KSmrq 08:28, 28 June 2007 (UTC)

Thank you for this integral. I will give it to my students tomorrow. =) I will also post other nasty ones I can find. –King Bee  12:04, 28 June 2007 (UTC)

How about simply

${\displaystyle \int {\frac {dx}{x^{3}+1}}?}$

The contrast between the complexity of the function and of the integral is greater. Arcfrk 15:53, 28 June 2007 (UTC)

Notation in improper integral limits

As it happens, I had experimented with various ways of indicating path of approach to zero in the different limits that are used, including "+" signs, diagonal arrows, vertical arrows, and nothing. In the end, I concluded that the least clutter was the most clear, which is what I used: just plain arrows ("→") and nothing more. This is the natural expectation, and I was careful never to approach from below so the reader never has to think to get it right. The fact is, many of our readers will have no prior exposure to the plus notation; it will not help them, but only get in their way. I'm convinced it's best omitted, and have acted accordingly.

Aside from that, I delighted to see that someone besides me is actually collaborating in editing on this "COTM", especially when they catch my silly copy errors. :-)

Only a few days are left in this month, and the list of suggested improvements has hardly been whittled down at all. I have a few more contributions in the works, and certainly the article is already better than it began. But I still find it an embarrassment for what should be a project showpiece. --KSmrq 11:25, 28 June 2007 (UTC)

This sets a bad precedent. The limit notation with pluses and minuses is standard, and we should report on it rather than skirt the issue. Further, you said that you were careful never to approach from below so the reader never has to think to get it right, but again, this is not a textbook where we are free to try our innovative pedagogic approaches. We should be up front about the issues, just avoid making them unnecessarily complicated. Arcfrk 16:02, 28 June 2007 (UTC)

I understand your dismay, but the suggestions at the top of the page are horrendously vague. "The computing integrals section could do with some expansion"? What does that even mean? We have main articles on all the techniques that one would learn in a calculus course. Should we have more examples? Use other, more advanced methods than the second fundamental theorem? Or should I just be bold and start going nuts? –King Bee  12:03, 28 June 2007 (UTC)

If you truly go nuts do it in the sandbox, see what happens--Cronholm 12:20, 28 June 2007 (UTC)

A-class review

An A-class review (a good way to rope in more editors) seems appropriate now that the end of the month is coming up, thoughts?--Cronholm 21:52, 28 June 2007 (UTC)

I think an A-class review would be premature. The article is struggling to be B-class, much less Bplus. For example, the content, organization, illustrations, and references (not yet one per section) all need work. Ask yourself, is it:

► correct? ► reasonably complete and balanced? ► clear? ► compelling? ► grammatical, correctly spelled, and well typeset? ► appropriately illustrated, where applicable? ► well linked? ► helpful in providing references and additional resources? ► reasonably accessible, given the topic?

Much has been accomplished; much remains. Imagine that your precocious young daughter becomes curious about integrals, and comes to Misplaced Pages for her first exposure. Is this the article you want her to see? Will she be oriented, educated, entranced, inspired? OK, so that's a bit much to strive for; but I know the article does not yet satisfy my A-class goals.

I still have a little time before the end of the month; I shall try to invest it well. --KSmrq 07:14, 29 June 2007 (UTC)

Well, the end of the month isn't going to be the end, it will be A class one way or another. :) --Cronholm 11:41, 29 June 2007 (UTC)

A precocious young daughter of mine getting interested in integrals? This sounds like a dream I once had; it was a good dream. =)

I will continue to think of ways to improve the article. –King Bee  13:15, 29 June 2007 (UTC)

The idea of me having a child of any age is frightening to me...--Cronholm 13:47, 29 June 2007 (UTC)

Sorry I haven't been more help of late (I had a burst early in the month), but I've been busy moving house. I do feel the article has been significantly improved, if nothing else then in terms of structure and fuller content for important sections. It does still need a lot of tweaking, and some conscientious referencing certainly wouldn't go astray; still, moving from start to class to B or B+ is not bad for month's effort. -- Leland McInnes 15:47, 29 June 2007 (UTC)

The lead

KSmrq, I'm afraid your edits are simply incorrect.

1. Refinement is not the correct word to use. Integration extends the concept of summation. Refinement refers to taking a subset, which is obviously not the case here.
2. Functions don't need to be continuous to be integrable, and that's that. Continuity is not a rquired condition in any definition of the integral, even the Riemann one. It's perfectly possible to integrate step functions. I suggest you take a look at Apostle's classic text Calculus.
3. The paragraph you removed was the only one that said anything about what an integral was. Unless you have a replacement, you should not remove it. Your lead does not tell the reader anything about what integration does in general.
4. "and dx denotes the weight in the "weighted sum", multiplying a height, say, into an area. (Introductory courses may treat dx as merely denoting the variable of integration.)" I don't know where you got this, but I can't find any mathematical meaning in it. Misplaced Pages is not the place for original research, please don't put in such strange statements unless you can source them. For a Riemann integral, dx does denote the variable of integration. It does so not only in introductory courses, but for graduate courses, doctoral students and professional mathematicians also. There's nothing anyone can do about it. In Lebesgue integral, it denotes a measure. Your statement is meaningless. Again, I suggest you take a look at Apostle.
5. Why do you insist on giving an incomplete statement of the fundamental theorem in the lead? Such auxiliary material does not belong in the introduction, they should go to the main body of the article. Loom91 07:09, 5 July 2007 (UTC)

I'm sorry your grasp of the English language and of integral calculus and of helpful writing is so limited, but please do not punish Misplaced Pages. You have repeatedly insisted on "correcting" things that are not mistakes. I'm afraid I will insist on reverting, and probably wasting my time explaining (as others have) precisely why.

1. The American Heritage Dictionary offers multiple definitions of "refine", none of which agree with you. One is "to acquire polish or elegance." Refinement is, of course, the act of refining; but also "a keen or precise phrasing; a subtle distinction." Ironic that you don't understand that. And were we to be using the word in a formal mathematical sense (which clearly is not appropriate here), you are still wrong; refining a partition for a Darboux sum is not removing a subset, to give one example.
2. This is now the nth time you have pulled this bullshit about continuity. Each time, the editor in question has not been saying what you assert in your reversion. Nobody has ever said we can only integrate continuous functions, nor only functions over continuous domains. We try to begin with something basic, and if you would look at a variety of calculus texts you will find that every author does so, as is only good pedagogy.
3. The removed paragraph said almost nothing; and what it did add, I incorporated as one sentence in the preceeding paragraph.
4. Again, it is you who needs to learn. Every definition of integration requires a combination of the function value and something more or less equivalent to measure, which is explicitly a "weight" in a "weighted sum". For the Riemann integral computing area under a curve, it comes from the width of the intervals in the Riemann sum. For the Lebesgue integral, say of a simple function, the measure is used precisely as a weight in a weighted sum. You seem unable to see what is right in front of you.
5. There are many variations of the fundamental theorem, and each has technical conditions. In fact, one of the motivations for defining different integrals is so that it can hold in greater generality. The lead is hardly the place to go into that. But mention of the fundamental theorem absolutely must go in the lead, as it is one of the most important facts about integrals that anyone should know.

That's a point-by-point rebuttal. And your insults are, well, a poor reflection on you. Enough. I'm reverting. (But I'll keep and correct your "Apostle" citation, which I had planned to add along with several others myself.) --KSmrq 12:36, 5 July 2007 (UTC)

I would like to mention tat the removed paragraph mentioned in point 3 does, indeed, say something, and would like to lobby for its re-inclusion, even if it is in a slightly cut down form. What I am looking for here is some material in the lede that provides a very accessible description of integration -- if you like a summary of the "Informal discussion" section. Currently the lede is quite concise, but fails to provide much for the truly mathematically naive reader -- something I think we can and should do, especially if it will only cost us a couple of sentences. -- Leland McInnes 13:47, 5 July 2007 (UTC)

I realized that's what the paragraph was trying to do, but it didn't succeed. For reference, here's what I removed:

---

Integration is used to find the "total amount" of a quantity that varies across a given domain. For example, the velocity of an accelerated body changes instant to instant through the continuous domain of time. To sum up all the instantaneous velocities over a given interval of time, and hence obtain the total displacement that occurred, we evaluate the integral of the velocity over the given interval of time. Though this concept was the starting point for the development of integration theory by Newton and Leibniz, it has since been extended, and newer definitions stress different aspects.

---

Despite repeated tinkering by different editors (including me), it ended up using a lot of words to little benefit. Frankly, I feel the same about the sprawling "Informal discussion". We absolutely do need friendly, gentle material early, but I'm not satisfied with what we've got. If someone else would be so kind as to fend off Loom91's misguided revisions, I can concentrate on my project of doing something about this, maybe with illustrations. If you like, I can go into more detail about my complaints, but I think my time is better spent rewriting so you can see what I'm trying for. --KSmrq 14:52, 5 July 2007 (UTC)

I agree that the tinkered with paragraph (it started out a little shorter I think) is not really quite what we want. I feel we should be improving it rather than removing it however. I would be more than happy to work with you on this -- one of my main interests in this article is ensuring it is accessible to a general audience. In that sense I would be happy with some discussion rather than just tinkering/changes. One of my major frustrations when first working on the lede was that people would simply tinker particular points that they wanted changed with little consideration for a coherent whole. I feel we woul do well to come to some agreement as to generally what we would like to say, along with ideas as to how best to say it, and draft it as a whole, rather than piecemeal tinkering at this stage.

With regard to the informal discussion: that is my work, though I still consider it somewhat of a draft. I spent some time working out generally what I wanted to say, and then sat down one afternoon and tried to write it as clearly as possible. The result, I admit, is a little sprawling, but I feel that the informal discussion is in many ways at least as important as any other, and deserves some space to decently communicate the underlying ideas. I would be happy to hear and discuss your criticisms of it (I have some myself that I haven't gotten to fixing yet) and hope that, with some discussion we can move toward something better. -- Leland McInnes 15:46, 5 July 2007 (UTC)

Sorry, can i just interject to ask where the grammar has gone on this article? A lot of the prose is very choppy, and doesn't flow AT ALL throughout the article. Also, it's become just like other articles where chunks of information are thrown onto the article and left. Please, tell me if there's any concensus to keep this way, otherwise i'd very much like to clean it up. ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ 15:31, 5 July 2007 (UTC)

I certainly would like to clean the whole article up for flow, but was waiting for it to stabilise on overall content a little first. Feel free to get started however. I certainly agree it could use some work. -- Leland McInnes 15:47, 5 July 2007 (UTC)

It would be a mistake to work on wording for flow just yet, when major sections may appear and move around. See below, for example, where we're discussing injecting a substantial section on complex integration. I'm going to be working over the section on Lebesgue integration, and some of the others are really more like rough drafts. If you look at the rating (top of this talk page), this is still just at start class, not even B, so it's considered a diamond in the rough, emphasis on rough. Thus if you come back in a week, big pieces of the article may look rather different. In my view, we're still at the grinding stage, and it would be a waste of time to start fine polishing. That said, I think there is more good content than two months ago.

What would be helpful now is to skim and report overall, and section by section: What do you notice that is missing, confusing, out of order, or possibly superfluous? Which sections are in decent shape, and which not?

For example, the history section has no business interjecting itself between formal definitions and properties, but there it sits — for the moment. The Lebesgue integral can't handle an improper Riemann integral discussed earlier on this page, which should be brought up in the Lebesgue section, except that improper integrals are currently much later in the article. To give two examples.

Did you just come late to the party? This was the Collaboration of the Month for June, and now you are kvetching? Ah well, better late than never, I suppose. And, honestly, much of the article hasn't been touched yet. --KSmrq 16:51, 5 July 2007 (UTC)

I will be honest here -- at the moment, it's not too pretty. A lot of the grammar even for experienced Mathematics majors would seem somewhat vague and nonspecific. Even all those years ago, before I went to university my tutors told me that the way that you should portray information if you want to be heard is to speak with clarity understood by everymen.

I still think it's a good idea to have a sufficient curve into the material to allow people who are unfamiliar with the subject to understand it, rather than for us to focus on the semantics of whether it's correct or not -- focus on the accessibility of the actual article for the majority of the accuracy, and we should just agree on a middle-ground for terminology.

I have free time tomorrow, so I'll work on it then. For now, let's just be careful with the article. ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ 20:31, 5 July 2007 (UTC)

The first and the last points are not important, let's talk about the other three.

• For a Riemann integral, dx denotes the variable of integration. I've provided a reference to the exact line of a textbook where this is explicitly said. I don't believe you have any source for the strange statement that it is a weight in a weighted sum.
• I assume by the most basic integral you mean the Riemann integral. A function does not need to be continuous to be Riemann integrable. Step functions are Riemann integrable (reference Apostle). If you are using some different definition of the most basic integral, please point that out. I've looked at several calculus textbooks, and none of them assert that a function needs to be continuous to be Riemann integrable.
• I don't know what sentence you are referring to. That paragraph may not have done a good job of giving an intuitive idea of what an integral is, but currently the lead does no job at all. An effort, even a poor one, is better than none. Write a better introducion, but untill you do, please don't remove the one that was existing. Loom91 12:56, 6 July 2007 (UTC)

Please stop this.

• Calculus books are written at many different levels for many different kinds of students. Collectively, they say all kinds of things, including falsehoods. Do you not know what a weighted sum is? Do you really need a textbook citation to believe that a Riemann sum (for example) is a weighted sum?!!
• If you understood integration more deeply, you would realize that the Riemann integral is critically dependent on a certain amount of continuity in the function itself; thus the Dirichlet function is not Riemann integrable. The first paragraph is not the place to go into technical details about isolated points of discontinuity; and the typical elementary examples are invariably continuous functions with continuous derivatives. You come up with a way to twist the interpretation of common language to dispute an unremarkable statement. Give it up.
• A quick click on the "Compare" button would answer your question. But since that seems to elude you, the sentence I added to the first paragraph (which also addresses your "continuity" nonsense) is "More sophisticated integrals can handle functions and domains of even greater generality."

You are simply making a nuisance of yourself, and not contributing productively to the development of the article. You have a habit of doing this, and have been warned about the same misbehavior at other articles. The fact that you misspell the well-known Caltech author "Apostol" as "Apostle" suggests that you are pretending to more expertise than you really possess, both in calculus and in English. So does the fact that you don't even bother to adhere to the article's Harvard style when you try to insert a reference. And the fact that you have written nothing to fill in missing sections of the article suggests you're really not interested in contributing. Therefore I suggest you go find something more productive to do with your time. (And by the way, I have written a better introduction, as well writing broad swathes of other content and producing numerous illustrations.) --KSmrq 13:58, 6 July 2007 (UTC)

Now you are violating several Misplaced Pages policies. You are engaging in revert war without participating in talk page discussion, you are being uncivil by calling edits you disagree with "bullshit", removing explicitly sourced content without providing rationale, and inserting unreferenced original research. All of these are against core Misplaced Pages policies. Please engage in civil discussion and provide references for edits. Loom91 13:30, 6 July 2007 (UTC)

You are mistaken. I took the time (just above you, if you will only look) to compose a thoughtful reply. You couldn't wait to read it. --KSmrq 14:02, 6 July 2007 (UTC)

In top of being uncivil, you are now lying. You wrote that reply long after you reverted me and inserted it between my two comments to make it appear as if I had not read your comment. This is not a good way of becoming a good editor. I'm afraid that the Misplaced Pages policy of providing references is not negotiable. If I provided a reference saying the Earth is square and you couldn't provide a source saying the contrary, my edit will stand. The fact is that I've provided a reference and you haven't. Stop reverting or I'm going to start a RfC. I assure you that other editors place even more emphasis than me on citing sources. Following policy, particularly core ones such as providing references and not doing original research, is not optional under any circumstances. What you or I do or do not believe has nothing to do with what stays in the article (and neither does expertise or qualifications, yours or mine). If you fail to agree with it, I'm afraid Misplaced Pages is not the correct place for you. May I suggest you carefully go through the Pillars of Misplaced Pages? A piecewise continuous function is not a continuous function. Making it so is not a simplification, it' an error. Your undoubtedly valuable contributions do not empower you to escape the policy of citation. Once and for all, provide references or cease this revert war. Loom91 16:17, 6 July 2007 (UTC)

I wrote my reply immediately, though it took some time. I took no steps to "insert it", but simply posted it in the usual fashion, immediately after your last remarks visible to me at the time.

I'm not impressed by your WikiLawyering. Describing a Riemann sum as a weighted sum is a trivial observation, and if you continue to attack it as demanding a citation you will only make yourself look (more) foolish. (For example, I note your arrogance towards Petergans, where you denigrate a university chemist with considerable expertise, and the way you attacked JRSpriggs' writing at Newton's laws.) And you persist in mischaracterizing the statement about continuity, in ever new ways, but always with your predetermined conclusion. You acknowledge my contributions, yet presume to lecture me. Your threats about citations are especially ironic in light of the fact that almost every citation in this article was put there by me, often with considerable digging to find the sources.

Meanwhile, you have yet to make any positive contributions to this article. Based on behavior, you have no interest in building an encyclopedia, only in making a nuisance of yourself. Well, congratulations; you're doing a fine job. --KSmrq 17:33, 6 July 2007 (UTC)

In an effort to bring some reasonable discourse back to this discussion... I've been thinking about how to provide some sort of intuitive/accessible description of integration for the introduction (really we only need a sentence, but something more would be very helpful). I wish to avoid "area under curve" descriptions because ultimately they are rather simplistic. Instead, going with the "weighted sum" description alredy present, I was hoping for something along the lines of:

Intuitively an integral may be thought of as a sum of an infinite number of infinitesimally weighted values.

The only sticking point with this for me is that "infinitesimally weighted" is perhaps a little too dense for an accessible description, something like ...of an infinite number of values, each given an infinitesimal weighting. One could raise objections about bringing infinitesimals into things, but they provide a more intuitive approach, albeit not a formally justifiable one (barring orking in smooth world toposes or similar). Thoughts? -- Leland McInnes 15:06, 6 July 2007 (UTC)

I agree that the current first paragraph is still less than ideal. We can try to improve it, but with so much major work pending for the rest of the article, I'd prefer to polish elsewhere just now. Note that article leads, especially first paragraphs, are popular battlegrounds. Why? Here's the implicit job description: "Describe Topic X in one sentence, which must be meticulously correct, totally complete, and easily understood by a great ape." :-) (Have you ever thought of what it would be like to write definitions for a dictionary? That's a real challenge!)

But since you ask, specificially about infinitesimals, I think that steps in a tiger trap. It's a technical detail about how integrals might be calculated. Too soon! What the reader wants to know in the first paragraph is, what is an integral, and why should I care?! From the inside (our view), your sentence is an answer to "what"; but from the outside (the lay reader's view), there is nothing intuitive about an infinite sum of infinitesimals. --KSmrq 15:39, 6 July 2007 (UTC)

I understand a certain reluctance with regard to infinitesimals, but I think we need to say something, and this represents a simple explanation. We need to say something about what an integral is in the lede, and currently there is nothing adequate there "sums of continuous functions over continuous domains" is far from satisfactory and the example is a touch technical with its terseness (you and I know full well what is meant, but it may well be less clear to those with, for example, a limited grasp of functions). I don't feel that the description given is "how integrals might be calculated", but rather a description of what simple integrals are -- you yourself are calling them "weighted sums", this merely expands on this to provide a description of a continuous/smooth weighted sum: one which sums infinitely many values, each of which is given an infinitesimally small weighting (since each represents an infinitesimally small part of the total sum). -- Leland McInnes 17:41, 6 July 2007 (UTC)

Contour integration/residue calculus

I have a question. Since use of the second fundamental theorem of calculus and the techniques one would learn in an undergraduate course are seemingly given a lot of consideration in this article, should we more thorougly cover the use of residue calculus to evaluate improper integrals? It could fit nicely in the improper integrals section, and I would be willing to write up an example and contribute images. However, if this is beyond the scope of what this article should contain (since this is a particular, albeit standard method), then I will not add it. I just want some opinions here. –King Bee  12:59, 5 July 2007 (UTC)

My initial concern is that this would result in the section on improper integrals being significantly larger than any other section, which certainly seems rather unbalanced under the circumstances. Some mention of residue calculus may not go astray, but I think we need to consider how best to work it in. Perhaps a short section on integration in complex analysis? -- Leland McInnes 13:40, 5 July 2007 (UTC)

Ah, it seems a few of us are still around feeling the need for more improvements. Good. My reading of the article is that it says very little technically about the fundamental theorem; we have a little overview, a little history, but nothing of substance. We also say almost nothing about complex analysis. So, I would like more of both. However, I agree with Leland that the improper integrals section is already large enough. It's already twice as long as the Riemann integral section (even though the latter now has a figure for convergence where I worked in a previous request to mention the Darboux integral)!

Singularities are important in theory, but to treat them properly we must work with complex integration. Why? Because a singularity in the complex plane, not on the real line, can cause problems for a purely real radius of convergence. And that gets us into complex analysis, which really deserves a section. So that's where you discuss residue calculus. --KSmrq 14:31, 5 July 2007 (UTC)

So, the impression I'm getting is that it would be a good idea to expand in this manner? I'm going to start whipping up some pictures tonight, and get going on some prose to accompany them. –King Bee  00:05, 6 July 2007 (UTC)

Yes, it is a Good Idea to introduce a section on complex integration, including a discussion of residue calculus.

I'm afraid that, so far, my contributions have not been paragons of economy; but the idea is a brief overview and orientation for each subtopic, leaving the bulk of the work to the specialized article.

Good luck with the pictures. I've seen some nice 3D depictions of poles and Riemann surfaces, but usually those take noticeably more work to produce. Before investing too much time, you might check Commons to see if there's anything we can appropriate. --KSmrq 14:37, 6 July 2007 (UTC)

Thanks for the advice. As far as the pictures were concerned, I was thinking something along the lines of examples of contours one might use to evaluate improper integrals of functions of a real variable. I've also been working on a better picture for the 3-D integration (rather than just the volume of a parallelpiped) that has a paraboloid or something like that. I should be making relatively sane progress throughout the weekend. –King Bee  15:19, 6 July 2007 (UTC)

One obvious place to look in Commons is Category:Mathematical analysis, but images are often poorly categorized. And the web is always a rich source for examples.

When making images, I would recommend keeping text to a minimum. A minor (though important) reason is to reduce clutter. A major reason is to increase reusability, both in other articles and in other languages. --KSmrq 16:00, 6 July 2007 (UTC)

State of the article

It was suggested that we should take an overall look at the shape of the article: the state of each section, what is missing, what is too long, etc. Let's get started on that. Fell free to add comments below.

1. Lede
   • Better than it was, but I feel it could use an eye to accessibility. Leland McInnes 19:49, 5 July 2007 (UTC)
   • The term "weight in the 'weighted sum'" is not defined or explained before. To me, it is rather distracting. At this point, I would only give the "introductory course" explanation, referring to the differential forms and measure sections for the actual explanation. (Most readers won't need to know this in the first place). The explanation of the terms integrand, domain of integration might be made shorter by something like

${\displaystyle \int _{\color {Blue}a}^{\color {Blue}b}{\color {Red}f(x)}\,dx.}$

Possibly these things might also be merged somehow with into a "notation" section (outside of history, though).Jakob.scholbach 23:48, 5 July 2007 (UTC)

1. Informal discussion
   • Still a draft, but contains most of the useful ideas; could use diagrams, and judicious cutting down. Leland McInnes 19:49, 5 July 2007 (UTC)
2. Formal definition
   • I'm reasonably happy with this, it could use some polish, but the core is in order. Leland McInnes 19:49, 5 July 2007 (UTC)
   • The section on other integrals should present for every type a short (one-phrase) description of the various types, otherwise it has more the character of a see-also-list. Several integrals are linked more than once. The phrase with the steel ball resting on a beam is incomprehensible (to me). A graph illustrating the intuition behind the Lebesgue-integral (something like this was discussed I remember) would be good (use a non-standard measure to illustrate Lebesgue's power!). Jakob.scholbach 00:06, 6 July 2007 (UTC)
3. History
   • Is a little out of place, but postponing it to the bottom was even worse. This is an important accessible section of the article, and burying it won't help non-technical readers. It needs to be up front somewhere. Content wise it covers the important points, though the post N&L section is light. Leland McInnes 19:49, 5 July 2007 (UTC)
4. Properties of the integral
   • I feel this hits the major points, but is a little terse. Concision is nice, but a little discussion wouldn't hurt. Leland McInnes 19:49, 5 July 2007 (UTC)
   • Concerning the basic inequalities: an image "proof" for the first given inequality would be nice (and easy to do and would not consume too much space). I miss the word "Schwarz inequality". The "Conventions" are IMO not a property, but rather a notation-related piece of information. Jakob.scholbach 23:48, 5 July 2007 (UTC)
5. Extensions
   • Is missing a discussion of integrals in complex analysis (which was in the original sandbox plan), and the multiple integrals subsection could really use some work. The rest is not bad, though the differential forms subsection could use some trimming down.
   • I disagree on trimming down the differential forms here. Rather I'd treat it as the first extension and try to explain or relate the other ones to diff. forms. The step from a "simple" integral to one with several variables (and also line integral-surface i.) is conceptually easier than differential forms. Therefore this deserves a pretty thorough discussion even in the integration article, I think. Jakob.scholbach 23:48, 5 July 2007 (UTC)
   • Where is non-classical analysis? --KSmrq 00:26, 6 July 2007 (UTC)

     Ugh, yes. We'll be getting awfully long if we do much more than skimming. I can write something on smooth infinitesimal analysis (I have Bell's text on this on hand at the moment) and could write soemthing to p-adic analysis, the rest is stetching beyond my field. -- Leland McInnes 00:36, 6 July 2007 (UTC)

     I think one sentence and the link would suffice; just say something. --KSmrq 04:05, 6 July 2007 (UTC)

     Perhaps its my soft spot for smooth infinitesimal analysis, but it may be worth saying a little more, since it provides such an elegant approach to integrals. -- Leland McInnes 04:52, 6 July 2007 (UTC)

6. Methods and Applications
   • Should probably come before Extensions; from a pedagogical perspective it is the easier of the two sections. In general I am unhapopy with the "Computing integrals" subsection, which is comparatively quite terse and lacking in detail. Leland McInnes 19:49, 5 July 2007 (UTC)

1. Missing Material
   • Integration in complex analysis, residue calculsu, etc. Should go in "Extensions" after multiple integration. Leland McInnes 19:49, 5 July 2007 (UTC)
2. Excessive detail
   • Nothing as yet, though the article length is getting long, so some cuts may have to be decided on in future. Leland McInnes 19:49, 5 July 2007 (UTC)

Introduction (was "Informal discussion")

As promised (threatened?), I did not make piecemeal changes to the informal discussion; I wrote a new one. It's even longer than its predecessor, and has a new name. However, I hope it makes up for that by saying more. Especially, I hope it is more gentle and helpful for the typical reader of this article.

It calls for two figures. The easier one shows two stepwise approximations to

${\displaystyle \int _{0}^{1}{\sqrt {x}}\,dx,\,\!}$

with five steps and right-end samples, and with seventy steps and left-end samples. These happen to be Darboux sums, so can be reused for that article. The harder picture is a 3D (or pseudo-3D) depiction of the swimming pool example, to illustrate integrals of volume, surface, and curve. I haven't yet thought much about its appearance.

But before I invest that effort, I'd like some feedback to know if the new section as a whole, or at least that example, is likely to be retained. Unfortunately, the response to calculus writing seems to vary widely, depending on the reader. (Read some reviews at Amazon for a taste of what I mean!) One reader may love it; another may hate it. The question is: will everyone hate it?

One last point: Leland McInnes invested time in writing the prior incarnation, and I promise I did read it more than once, and did try to draw inspiration from it. I used none of those words and none of those examples, but the good in it lives on. If that was a first draft (as was said), then this is a second draft.

Enjoy. --KSmrq 15:15, 6 July 2007 (UTC)

I'm working on your sqrt(x) picture right now. Should I keep the theme with that cool blue color you have thrust forward (for the shading of the rectangles), or should I switch it up? –King Bee  15:51, 6 July 2007 (UTC)

Actually, I've got it in progress, using the same machinery I used for the previous illustrations. (I would have it done already, if I'd stop talking!) My originals are SVG; unfortunately the software MediaWiki has chosen is a painfully deficient implementation, so I oftne upload PNG instead. But for reference, you can find my standard four colors at the bottom of my user page.

These are not arbitrary choices; they are experimental results from studies of the human visual system, so-called "unique hues". As data makes its way from the retina to the brain, it is encoded using an opponent process. The most important axis is black/white; it also has the highest resolution. Next is red/green; and last is blue/yellow, which has noticeably lower resolution. This influenced the design of the analog television upgrade from black-and-white to color. It also shows up in the Berlin and Kay investigation of basic color terms in languages around the world. So my blue is a really special blue, not just a tasteful choice. :-) --KSmrq 16:38, 6 July 2007 (UTC)

Wow, and here I thought it was just a really pleasant shade of blue. =) (As a note, the other picture I contributed used the same hexcode as your particular shade.)

Well, if you have the picture in progress, maybe I'll just let it be, and work on other pictures. I also create images in inkscape, and they are svg; and I upload as png for the same reason you described. –King Bee  17:47, 6 July 2007 (UTC)

I have to admit to being a little unhappy with this new version; in part it is a little long, and in part I feel it fails to highlight what I feel is the important distinction of integration, which is resolving the issue of the continuous. I'll try and proviode more detailed comments on this, and suggestions for improvement, at a later date. -- Leland McInnes 18:26, 6 July 2007 (UTC)

I am more than a little unsatisfied with the introduction (formerly, informal discussion). The main problem seems to be the textbook tone it takes, together with the substantial increase in sophistication after the last revision. Remember: the goal of this section is not to summarize what we know about the integral! That honour belongs to the lead. Rather, it should be a gentle motivation for integration and integrals, if indeed that is possible. In particular, any types of numerical examples, and especially, worked out problems, appear extremely unencyclopaedic, to the point of violating WP:NOT#TEXT. Looking over the recent attempts at the informal introduction, I've begun to think that, perhaps, including it may not be a good idea, at least, if we want this entry to read as an article in an encyclopaedia. Opinions? Arcfrk 02:00, 7 July 2007 (UTC)

I feel that something in the way of an informal discussion is important for WP:MTAA. Integral is an important article and should be one of the foundation articles regarding calculus, thus it should provide material for interested readers with no understanding of even basic calculus concepts, ideally as close to the beginning as possible. Technical material can be deferred till later in the article. I certainly agree that the aim should not be to summarise everything we know about integrals. The aim should be to provide some semblance of discussion of integrals for a general audience. I felt my earlier version was too long and could be usefully cut down to the short discussion required. I feel the current version is far too long and covers far too much material. Given the nature of the revised version I feel we should perhaps start with some discussion of exactly what we would wish for the section to communicate, since apparently I and KSmrq have rather different ideas as to what should be discussed. -- Leland McInnes 02:26, 7 July 2007 (UTC)

The first problem I perceieve with the current introduction (formerly "informal discussion") is that it is just too long, often taking much time to discuss topics beyond its scope. For example, the second paragraph is long and essentially purely about the issue of incommensurable lengths, and the real number line. While some discussion is useful in that addresses the reasons for the early incompatability of discrete and continuous worlds, it is well off topic about integrals.

The second point I want to discuss is the amount of semi-worked examples. This is not a textbook, and worked examples are not required; in practice they simply extend the length with the necessary explanations of particulars.

My third issue is with the relative level of technicality, and the degree of mathematical sophistication required. While the sesction begins with a very general tone, it is not long before we are blithely tossing around functions with little or no explanation. If this section is to be a description for general audiences that I feel it should be we should be following the Hawking view that "every equation halves the readership" and try keep things at a low level. When the introduction continues and gets into Lebesgue integrals and differential forms I feel we are extending well beyond what can be adequately covered in such a section.

My essential point is that it seems that I and KSmrq have rather different ideas about what this section should be. This probably needs to be fixed before we can proceed. Towards that end, let me make clear my own views on what the section should contain so that we can work toward some consensus or compromise position. I feel the role of this section is to provide a relatively concise description of what integration is for a completely general audience. That is, where we should provide a one sentence "intuitive" description of integrals in the lede, this section should be nothing more than the expansion of that sentence to a paragraph or two. In short it should be the informal definition (being very informal and high level).

What specifically do I feel the section should discuss?

• The opposition of the discrete and the continuous.
• A high level view of how integration extends/refines discrete sums into the realm of the continuous.
• Some manner of physical example showing how this is useful; be it areas, or rates of change -- the key point being that it deals with continuous worlds (geometry, or time) rather than discrete ones.

I don't feel anything more is required. Equally, I feel that these points should be addressed as directly as possible.

I hope this will help us move forward in discussion of how to handle the "introduction" (formerly the "informal discussion"). -- Leland McInnes 17:26, 7 July 2007 (UTC)

• I'm quite happy with the introduction as it is now. First, I think it is not too long. The subject is not small enough to give a short intro without cutting essential parts of the story. IMO this is exactly the place where a comparing glance over the different types of integration should take place (as it is done right now). A reader with no or little knowledge of calculus needs to be able to "understand" the part up to the Darboux sums, which I think he could. A high-school will read until he meets his friend(?) "Fundamental theorem of calculus". I would add "the related function F(x), the so called antiderivative", though. Finally, a undergrad/grad student will be able to read up to Lebesgue and Stokes. So all kinds of readers find a piece of information fitted to their level/interest.
• I agree with the previous poster that the opposition of discrete and continuous is important - it seems to be there, namely in the "In the simpler case where y = 1, the region under the 'curve'". However, a different wording may enlighten the intention more. Perhaps compare summing sqrt(0)+sqrt(1)+sqrt(2)+sqrt(3)+sqrt(4) to the integral of the function from 0 to 4. Even better would be to find some physical phenomenon (like an accelerating rocket or so, but this is pretty complicated, I remember faintly) - having a simple functional description with simpe antiderivative and then discuss this function. This would also make obsolete the existence of two different examples.
• The only point which I would drop is the one on the Pythagorean theorem. Probably a reader willing to learn about the integral will know that the reals are a continuum. Even if not, when one starts out with a physical example, it should be intuitively clear to anyone that it is not enough to sum up the speed of the rocket at 1,2,3,4 seconds after take-off.
• In the last subsubsection on Stokes, the f(x)dx is not what corresponds to the differential form ω, rather it is the antiderivative F(x). By mentioning above that F is called antiderivative, this can be included here without further ado. Jakob.scholbach 17:03, 7 July 2007 (UTC)

Request for comment

I and KSmrq are currently locked in dispute about the lead section of the article diff between preffered versions. As you can see, I've provided a precise source for my version. KSmrq has reverted it repeatedly without bothering to provide a counter-reference. On top of that, he has repeatedly engaged in incivil behaviour, such as calling me as a nuisance and lying to portray me in an unfavourable light . I request the community to look into the matter. Thank you. Loom91 16:43, 6 July 2007 (UTC)

Comments on the nature of this dispute can be found here, including some advice and policy reminders from mediators. Discussion pertaining to article content should remain on this page. — KieferSkunk (talk) — 01:29, 7 July 2007 (UTC)

Discussion

I can somewhat sympathise with both sides of this debate, but I've been trying to stay out of it because it has/is rather heated (for reasons that are beyond me -- it seems to me sensible compromises are available). The following are my own views on the various points Loom91 wishes to dispute.

1. This seems a minor point, and can be quibbled about later. Refinement seems fine to me.
2. I understand KSmrq's view that in referring to basic/simple integration it is continuous functions that are considered. However, it is adding an extra, and in practice unnecessary, qualifier to "function". The important issue is that of the continuity of the domain; I don't feel that referring specifically to continuous functions makes things at all easier to understand for naive readers. It's probably easier to leave it out.
3. This point I'm already discussing. I understand the desire to remove the paragraph, and I am working to try and build a suitable replacement. I see no problems here presuming discussions do not get roadblocked for some reason, so this point is moot.
4. dx does denote a weight, and can be read that way. Indeed, that was its use in Leibniz original notation -- it denoted an infinitesimal weight. Extending a little, with differential forms the basic forms do indeed denote densities which weight the various functions. There is some avoidance of this view in introductory texts because of the fear of infinitesimals (which it originally denoted) but I wouldn't say that it is wrong, especially given the differential forms, non-standard analysis, and smooth infinitesimal analysis approaches. Can we put both views on equal footing (rather than demoting one to being parenthetical, or removing it altogether) with something like "dx may be alternately viewed as either ... or ..."?
5. Postponing tFToC till a proper discussion can be made, or promoting it to be more fully discussed earlier may be of benefit. This is more of a minor quibble as far as I can tell.

Hopefully this is helpful and can help cool the debate somewhat -- there are reasonable compromises here, so hopefully cool heads can prevail? -- Leland McInnes 18:45, 6 July 2007 (UTC)

I will like to point out that that was exactly what I had originally done, mentioning that dx can have different interpretations depending on the theory and giving three examples. KSmrq's version gave the impression that dx was treated as the variable of integration only in simplified texts for beginners, when it is a perfectly valid interpretation at all levels. Loom91 07:13, 7 July 2007 (UTC)

The phrase on what "dx" denotes seems to be the main conflict area now. It can most definitely be viewed as denoting a weight; it corresponds to the Δ_i mentioned in the section on the Riemann integral. However, I think that most introductory calculus texts treat it as formal notation. The best way to treat it in this article depends on how the rest is written, but for the moment I agree with Leland's proposal so I implemented it. I removed the reference to the book by Dey because I can't find any information about it, suggesting that it is rather obscure. -- Jitse Niesen (talk) 19:37, 6 July 2007 (UTC)

I think treating it as a weight smudges the fine but important distinction between a Riemann sum and a Riemann integral. Loom91 07:13, 7 July 2007 (UTC)

The language of the intro is far too informal, anyway, so I don't see any reason to bicker over this small point. It's not the job of the intro to explain calculus to 4th graders. For comparison, here is Britannica's entire article on "integral":

in mathematics, either a numerical value equal to the area under the graph of a function for some interval (definite integral) or a new function the derivative of which is the original function (indefinite integral). These two meanings are related by the fact that a definite integral of any function that can be integrated can be found using the indefinite integral and a corollary to the fundamental theorem of calculus. The definite integral (also called Riemann integral) of a function f(x) is denoted as

(integral f(x)dx)

(see integration ) and is equal to the area of the region bounded by the curve (if the function is positive between x = a and x = b) y = f(x), the x-axis, and the lines x = a and x = b. An indefinite integral, sometimes called an antiderivative, of a function f(x), denoted by

(integral f(x)dx)

is a function the derivative of which is f(x). Because the derivative of a constant is zero, the indefinite integral is not unique. The process of finding an indefinite integral is called integration.

Just my 2 cents. Gnixon 22:27, 6 July 2007 (UTC)

It's nice that Britannica is that concise, however I think the relevant points here are WP:NOTPAPER and WP:MTAA. We have entirely separate articles for Riemann integral, Lebesgue integral and Henstock-Kurzweil integral, as well as Antiderivative and more. Those articles can handle greater tehnicality as required (though we certainly delve into the technicality in this article). I think it important that, as a core article on calculus, this article be accessible to a general audience. Sure, the "formal definition" and later sections can provide detail that a general reader may not follow, but we should provide sufficient general material up front to provide an average interested reader with at least an understanding of the core concepts/ideas of integration. -- Leland McInnes 02:34, 7 July 2007 (UTC)

This is very important. The introduction needs to introduce; it is the most important part of the article for majority of readers coming to the page and needs to serve a non-mathematician reader (adult, not fourth-grader) as well as the mathematical. It should not, for instance, use the (at that point) undefined term "Riemann integral". -R. S. Shaw 05:22, 7 July 2007 (UTC)

New lead

I have written a new lead, which summarizes the content of the article more closely, as recommended by WP:LEAD. I feel that the technicalities (continuous vs general functions, area vs signed area, description of the Riemann sum) should be defered to the main text. In particular, in the interest of clarity, I suggest replacing the present picture of a sign-changing function with a picture of a positive function. Arcfrk 06:14, 7 July 2007 (UTC)

I like the current lead. But while it discusses the Riemann integral in some detail, the general purpose of an integral, that of totalling, is not really alluded to, nor is its relation to a sum. I think the examples of various types of integrals should be replaced by some general discussion of why anybody wants to use an integral. Anyway, my purpose of initiating this RfC has suceeded: peace and sensibility has been restored. Thanks to everyone who helped out.Loom91 07:21, 7 July 2007 (UTC)

I agree with Loom91 that the current lede lacks an accessible description, which the previous lede offered. On the other hand I feel this is, perhaps, symptomatic of the current state of the "introduction" which I feel fails to provide the accessible description of integration that it should; were that fixed then the lede could suitably summarise that section and all would be well. I'll continue trying to discuss reform of the "introduction" section and hope this will clarify lede issues. -- Leland McInnes 14:18, 7 July 2007 (UTC)

As pointed out previously by Leland McInnes, part of the difficulty here is the sprawling nature of this article. It would certainly be nice to have a one or two-sentence description of integration as a continuous version of summation. On the other hand, we should try to avoid focusing too much on details of the Riemann integral, since it is treated in a separate article, with its own lead, introduction, and explanations. I was not able to come up with anything that at the same time adequately reflected the idea of integration and were concise enough and gentle enough to warrant inclusion in the lead. If anyone can accomplish it, please, do! Arcfrk 19:46, 7 July 2007 (UTC)

• Former good article nominees
• Start-Class mathematics articles
• Top-priority mathematics articles
• Misplaced Pages pages with to-do lists