Talk:Integral/Archive 2

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Assessment comment

The comment(s) below were originally left at Talk:Integral/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

* This article needs to be expanded to cover the topic more broadly. Integration is one of the key mathematics articles, and it should provide a good balanced coverage of the concept. Article with this name should not be reserved for only Integral of real-valued functions one real variable. See talk page for more suggestions. Stca74 17:24, 15 May 2007 (UTC)

Last edited at 11:39, 2 June 2007 (UTC). Substituted at 14:32, 14 April 2016 (UTC)

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)