Talk:Integral

Antiderivatives

Resolved

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) [the integral] 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

[F(x + h) - F(x)]/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: [F(x + h) - F(x)]/h = f(c)

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

Now as h --> 0 c will approach x

[F(x + h) - F(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^T 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 [a, b] 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

Resolved

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 Wikipedia, 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 [a,b].

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 $\int f(x)dx$ while others write $\int dxf(x)$ . Is there a story behind these two different conventions? --Itub 13:21, 21 May 2007 (UTC)

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

I don't think that both ways are good. $\int f(x)dx$ is good, but $\int dxf(x)$ means $\int 1dx*f(x)$ 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

Resolved

Is this always true: $\int _{a}^{b}f(x)\,dx=-\int _{b}^{a}f(x)\,dx$? --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¢

Resolved

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

Resolved

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^T 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 $x^{x}$ 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^T 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--Cronholm¹⁴⁴ 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 Wikipedia: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 Wikipedia. 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 $=F(b)-F(a)$, 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^T 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.--Cronholm¹⁴⁴ 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^T 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^T 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 ^{(τ • γ)} 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^T 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^T 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? --Cronholm¹⁴⁴ 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
$\int _{a}^{b}f(x)dx=\lim _{c\to a}\int _{c}^{b}f(x)dx$
(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 ^{(τ • γ)} 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^T 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^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 ^{(τ • γ)} 18:33, 24 June 2007 (UTC)

(editconflict)

X^X falls under Wikipedia: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--Cronholm¹⁴⁴ 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. --Cronholm¹⁴⁴ 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^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^{−log(x+2)/2}, 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^T 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 $e^{-|x|}$ 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 ^{(τ • γ)} 02:09, 25 June 2007 (UTC)

Correct. It is trivial to do one or the other; x^−p 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^T 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

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

Then the derivative of F exists and is finite in [0,1], but is not Lebesgue-integrable in [0,1]. For, let

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

so that

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

The intervals [a_n,b_n] are pairwise disjoint; thus if E is their union, then

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

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^T 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 $F'$ (the function discussed above) does not exist, the Riemann integral does; it is equal to -1. –King Bee ^{(τ • γ)} 12:48, 26 June 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):

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

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 ^{(τ • γ)} 19:16, 25 June 2007 (UTC)

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

Glad to help out. –King Bee ^{(τ • γ)} 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 ^{(τ • γ)} 12:44, 26 June 2007 (UTC)

  Quick answer: Done. --KSmrq^T 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^T 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 ^{(τ • γ)} 11:14, 27 June 2007 (UTC)

The derivative of arctan(u) where u is equal to x^-1/2. He skipped a bunch of steps.--Cronholm¹⁴⁴ 11:39, 27 June 2007 (UTC)