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Should we add a section on the multivariate integral before the standard integral in one dimension?
I wonder what readers will be better served by
having a detailed section on integration in higher dimensions,
before the article even discusses the basic one-dimensional case. This doesn't seem likely to help the intended readership of this article. Extensions to higher dimensions, line integrals, and surfaces integrals are already covered in their
own section. I don't see how adding a bunch of duplicate content to the top of the article is likely to enhance the readability of the article. I'm willing to be proven wrong, but ideally the role for such content, and why the article should be restructured this way, should be discussed. Edit-warring is unconstructive, because of
WP:BRD. I've tried to improve the recently-added content, in the spirit of
WP:CON, but discovered in doing so how little really worthwhile content there was. So, please don't revert. It's time to discuss!
Sławomir
Biały
21:01, 12 November 2015 (UTC)
I have never seen it used in practice. Do you have references? J.P. Martin-Flatin ( talk) 11:32, 13 November 2015 (UTC)
The lead should stand on its own as a concise overview of the article's topic. It should define the topic, establish context, explain why the topic is notable, and summarize the most important points, including any prominent controversies.[1] The notability of the article's subject is usually established in the first few sentences. The emphasis given to material in the lead should roughly reflect its importance to the topic, according to reliable, published sources. Apart from basic facts, significant information should not appear in the lead if it is not covered in the remainder of the article. As a general rule of thumb, a lead section should contain no more than four well-composed paragraphs and be carefully sourced as appropriate.
No, I don't think we should reduce it to the point where it no longer communicates anything. For example, your recent edit to the section on differential forms is now pretty much incomprehensible to likely readers of this article. A paragraph or two is fine for summary style. See, for example, the mathematics good articles
Hibert space or
Group (mathematics) for examples of how summary style works. The surface integrals and line integrals sections seem about right to me now.
Sławomir
Biały
14:10, 14 November 2015 (UTC)
In its current form, this article is very long, and its scope is a bit blurred by the fact that we start from rock bottom up to exterior derivatives and symbolic integration. As a result, the target audience of this article is unclear and we may raise expectations far too high. I think we need to reduce the scope of the article and shorten it, to set readers' expectations at the right level.
Starting with the low-hanging fruits, I would like to transfer all the material currently in section " Computation" into a new article called "Computation of integrals", keeping only a very short summary here and a pointer to that new article. What does the community think about it? Is there a majority in favor of this change?
In the previous section, User:Slawekb suggested to limit the scope of this article to integrals over a real interval, which would also help tighten the scope and set expectations right. I leave it to him to handle this change, which requires much material to be deleted from section " Extensions" and may raise some opposition. J.P. Martin-Flatin ( talk) 14:39, 13 November 2015 (UTC)
Examples:
Steven Weinberg, The quantum theory of fields.
Raymond Paley and
Norbert Wiener Fourier transforms in the complex domain.
Richard Courant and
David Hilbert, "Methods of mathematical physics" (see, e.g., volume 1, section II).
Sławomir
Biały
17:32, 14 November 2015 (UTC)
Area under the curve redirects here, which is somewhat confusing for people who are looking for Area under the curve (pharmacokinetics). A hatnote I placed here has been reverted. Any objections if I turn Area under the curve into a disambig? Or are there better solutions? Thanks -- ἀνυπόδητος ( talk) 10:49, 15 December 2015 (UTC)
"Integral calculus"(en) redirects to this page, "Integral"(en), which links to the german "Integralrechnung"(de). Only "Integralrechnung"(de) doesn't link back to "Integral"(en). By the way: "Integralrechnung" means "integral calculus". Could this be fixed in some way? Téleo ( talk) 08:40, 13 January 2016 (UTC)
The following addition to the history section of article looks to me as propaganda from some Indian nationalist - it's without reliable citation etc. A swift action be taken in this regard.
In India around 15th century, in the Jyeṣṭhadeva veda, we find the notion of integration, termed sankalitam, (lit. collection), as in the statement:
(SarfarazLarkanian 19:56, 7 March 2016 (UTC))
The diagram to illustrate the Lebesgue integral is a howler. Lebesgue's original idea is to divide the range of the function into interval, but that does NOT mean that the area below the graph is divided into horizontal strips. Instead, the intervals are projected down onto the x axis. Most texts no longer use that approach. Instead they approximate the function by simple functions. The crucial difference between the Riemann and Lebesgue integrals is that the latter multiplies the value of the function in an interval by the MEASURE of its projection onto the x axis. TerryM--re ( talk) 12:01, 16 April 2016 (UTC)
It has been asserted several times in this discussion that the definition given in the article does not agree with Lebesgue's own definition. One of Lebesgue's definition was as follows (refer to the first two paragraphs appearing in section 5.3 of the aforementioned book by Williamson), for a bounded non-negative measurable function f on a measurable set E, with . Fix an and an integer N such that . For , , let denote the measurable set
Let and . Then , and the Lebesgue integral is the common value of and .
This is related to the definition given in the article as follows. Let . For each positive integer N, let denote the partition of the range of f given by . Let and denote the upper and lower Darboux sums for approximating the integral from the article. The supremum of for t in an interval is at most , and the infimum is at least , so that, by definition, we have and .
This proves that the definition given in the article is equivalent (in a fairly trivial "from the definition" way) to the Lebesgue approach. In other words, Lebesgue's definition of the integral really is trivially just given by the Riemann-Darboux integral of the distribution function . Hopefully this lays all further objections to rest. Sławomir Biały ( talk) 11:54, 21 June 2016 (UTC)
While this article is very long (and rightfully so), I believe it needs an "Applications" section. Otherwise the uninitiated reader will ask "What's the point?" To avoid having the applications section go unnoticed by readers who may not look far down for it, I'm going to follow the WP standard and put it early in the article since it should be as elementary as possible, and since our math articles start out simply before getting more complicated. Expansion of the section, while keeping it simple, would be welcome! Loraof ( talk) 20:11, 7 July 2017 (UTC)
Why can't I write about the usage of the integral in Kinematics or integrals of standard functions? There is no separate article for it. Should I create a new article? The reverse power rule and stuff like that is one the first things you learns in integral calculus. Lie Cleaner HK 17:32, 15 August 2017 (UTC)
The addition of the isn't needed here. This is a common abuse of notation that's generally understood from context. Moreover, the explanation is really getting out of scope for this article, especially for the lead since this the article is about integration, and not antidifferentiation. A footnote might be okay here, but that's still probably overkill. -- Deacon Vorbis ( talk) 16:53, 15 August 2017 (UTC)
Take a look here: /info/en/?search=Constant_of_integration#Necessity_of_the_constant Lie Cleaner HK 17:32, 15 August 2017 (UTC)
I apologize. I realize now that the +C is implied in the indefinite integral. I confused this with a more strict formula for the indefinite integral containing an initial condition. ScaAr ( talk) 21:30, 29 August 2017 (UTC)
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Much of it seems like an essay which is not good encyclopedic practice at least with the high standards of english wikipedia. — Preceding unsigned comment added by Yoandri Dominguez Garcia ( talk • contribs) 14:50, 15 June 2018 (UTC)
@ Rgdboer: You say "in-line references are not appropriate". Are you quoting the Wikipedia Manual of Style or Mathematics guidelines? I have looked in those documents and found nothing about in-line references being inappropriate in articles on the topic of mathematics, or less valuable than articles on other topics.
Over the years, nine Math articles have been raised to Featured Article status. Those nine, and their number of in-line citations are as follows:
There may be some truth in the idea that articles about Mathematics do not require as many in-line citations as articles on other topics. If that is so, the Mathematics guidelines should be amended to make it clear and put the matter beyond dispute. Until then, I think User:Yoandri Dominguez Garcia is justified in drawing our attention to the fact that this article does not meet Wikipedia’s standards regarding in-line citations. Dolphin ( t) 11:18, 30 October 2019 (UTC)
@ Deacon Vorbis: The reversion of my contribution concerning hyperbola quadrature and its place in the History of Integration suppresses useful information. "Unreferenced" says the edit summary, yet Introduction to the Analysis of the Infinite is part of the contribution. Although it is an Original Source there is an article on that book from 1748, and our article has several secondary sources, including Henk Bos who concurs with what was contributed here. Also, referring to History of logarithms there is a statement by D. T. Whiteside that supports the contribution. This material is very old so accusations of OR such as you raised July 18 this year in Talk:Natural logarithm are bogus. The time you put in to improve this Project is appreciated; however, it appears you are repressing information about the century (1647 to 1748), before natural logarithm, when hyperbola-quadrature was used. Why not tag the contribution with "citation needed" rather than revert. Restoration of the contribution is requested by this Talk, perhaps citing Whiteside. Please respond since dialogue here is required before appeal to a higher venue. — Rgdboer ( talk) 20:39, 30 October 2019 (UTC)
{{
math}}
/{{
mvar}}
, and some wasn't even italicized), but that can be fixed. There were also some problems with the style of the prose; that's more difficult to fix without having a source to refer to – another important reason to include them. For just one example, you said, "Promulgation of the hyperbola-quadrature by Huygens and Nicholas Mercator assured the transcendental function's acceptance."This is kind of confusing, and I, for one, wouldn't be able to clear it up without having a source to refer to. A claim such as this really needs a source anyway. – Deacon Vorbis ( carbon • videos) 21:02, 30 October 2019 (UTC)
I don't believe that "Signed Area" should redirect to this "Integral" page. When I hear "signed area" I don't think only of the "areas are negative below the x-axis" convention for integrals. I also think of the related-but-more-general concept of, say, considering areas of regions enclosed by counter-clockwise paths as positive and clockwise paths as negative. This comes up very naturally when considering, say, the Shoelace formula. -- Helopticor ( talk) 13:05, 19 April 2020 (UTC)
This is a small point. I spotted a mistake in the definition of the Riemann integral, which included the following segment:
This is a typical argument of the epsilon-delta type. The mesh of a partition is the width of the largest sub-interval formed by the partition. If the width of the largest sub-interval (with some index k which we don't need to know) is , this implies that for all sub-intervals are . No need to go at the level of indices or of taking into account the plurality in the notation: the notion of mesh does the job.
So the correct formulation should be (and using lower case delta makes the argument even clearer, showing that it is the familiar epsilon-delta argument):
I will try again, asking all those who want to revert my change to read the above comment and indicate where it goes wrong, if you find something wrong with it.
Dessources ( talk) 13:13, 15 August 2021 (UTC)
Finding area by integration on the area between curve y = f(x) and x-axis? — Preceding unsigned comment added by 129.232.97.252 ( talk) 16:56, 14 May 2022 (UTC)
[2] breaks out separate sections for analytical vs symbolic integration, but I was raised that analytical and symbolic mean the same thing in this context. Is there some different meaning I'm not aware of? Rolf H Nelson ( talk) 04:56, 4 January 2022 (UTC)
I also agree that the article is muddled: finding an antiderivative is described in both the "Analytical" and "Symbolic" subsections. The article can be improved.
I am trying to improve the lead sentence since it came up in the village pump as an example of something that needs work. My contribution is based on the suggestions from a WikiProject:Mathematics discussion Thenub314 ( talk) 16:19, 10 February 2023 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 3 | Archive 4 | Archive 5 |
Should we add a section on the multivariate integral before the standard integral in one dimension?
I wonder what readers will be better served by
having a detailed section on integration in higher dimensions,
before the article even discusses the basic one-dimensional case. This doesn't seem likely to help the intended readership of this article. Extensions to higher dimensions, line integrals, and surfaces integrals are already covered in their
own section. I don't see how adding a bunch of duplicate content to the top of the article is likely to enhance the readability of the article. I'm willing to be proven wrong, but ideally the role for such content, and why the article should be restructured this way, should be discussed. Edit-warring is unconstructive, because of
WP:BRD. I've tried to improve the recently-added content, in the spirit of
WP:CON, but discovered in doing so how little really worthwhile content there was. So, please don't revert. It's time to discuss!
Sławomir
Biały
21:01, 12 November 2015 (UTC)
I have never seen it used in practice. Do you have references? J.P. Martin-Flatin ( talk) 11:32, 13 November 2015 (UTC)
The lead should stand on its own as a concise overview of the article's topic. It should define the topic, establish context, explain why the topic is notable, and summarize the most important points, including any prominent controversies.[1] The notability of the article's subject is usually established in the first few sentences. The emphasis given to material in the lead should roughly reflect its importance to the topic, according to reliable, published sources. Apart from basic facts, significant information should not appear in the lead if it is not covered in the remainder of the article. As a general rule of thumb, a lead section should contain no more than four well-composed paragraphs and be carefully sourced as appropriate.
No, I don't think we should reduce it to the point where it no longer communicates anything. For example, your recent edit to the section on differential forms is now pretty much incomprehensible to likely readers of this article. A paragraph or two is fine for summary style. See, for example, the mathematics good articles
Hibert space or
Group (mathematics) for examples of how summary style works. The surface integrals and line integrals sections seem about right to me now.
Sławomir
Biały
14:10, 14 November 2015 (UTC)
In its current form, this article is very long, and its scope is a bit blurred by the fact that we start from rock bottom up to exterior derivatives and symbolic integration. As a result, the target audience of this article is unclear and we may raise expectations far too high. I think we need to reduce the scope of the article and shorten it, to set readers' expectations at the right level.
Starting with the low-hanging fruits, I would like to transfer all the material currently in section " Computation" into a new article called "Computation of integrals", keeping only a very short summary here and a pointer to that new article. What does the community think about it? Is there a majority in favor of this change?
In the previous section, User:Slawekb suggested to limit the scope of this article to integrals over a real interval, which would also help tighten the scope and set expectations right. I leave it to him to handle this change, which requires much material to be deleted from section " Extensions" and may raise some opposition. J.P. Martin-Flatin ( talk) 14:39, 13 November 2015 (UTC)
Examples:
Steven Weinberg, The quantum theory of fields.
Raymond Paley and
Norbert Wiener Fourier transforms in the complex domain.
Richard Courant and
David Hilbert, "Methods of mathematical physics" (see, e.g., volume 1, section II).
Sławomir
Biały
17:32, 14 November 2015 (UTC)
Area under the curve redirects here, which is somewhat confusing for people who are looking for Area under the curve (pharmacokinetics). A hatnote I placed here has been reverted. Any objections if I turn Area under the curve into a disambig? Or are there better solutions? Thanks -- ἀνυπόδητος ( talk) 10:49, 15 December 2015 (UTC)
"Integral calculus"(en) redirects to this page, "Integral"(en), which links to the german "Integralrechnung"(de). Only "Integralrechnung"(de) doesn't link back to "Integral"(en). By the way: "Integralrechnung" means "integral calculus". Could this be fixed in some way? Téleo ( talk) 08:40, 13 January 2016 (UTC)
The following addition to the history section of article looks to me as propaganda from some Indian nationalist - it's without reliable citation etc. A swift action be taken in this regard.
In India around 15th century, in the Jyeṣṭhadeva veda, we find the notion of integration, termed sankalitam, (lit. collection), as in the statement:
(SarfarazLarkanian 19:56, 7 March 2016 (UTC))
The diagram to illustrate the Lebesgue integral is a howler. Lebesgue's original idea is to divide the range of the function into interval, but that does NOT mean that the area below the graph is divided into horizontal strips. Instead, the intervals are projected down onto the x axis. Most texts no longer use that approach. Instead they approximate the function by simple functions. The crucial difference between the Riemann and Lebesgue integrals is that the latter multiplies the value of the function in an interval by the MEASURE of its projection onto the x axis. TerryM--re ( talk) 12:01, 16 April 2016 (UTC)
It has been asserted several times in this discussion that the definition given in the article does not agree with Lebesgue's own definition. One of Lebesgue's definition was as follows (refer to the first two paragraphs appearing in section 5.3 of the aforementioned book by Williamson), for a bounded non-negative measurable function f on a measurable set E, with . Fix an and an integer N such that . For , , let denote the measurable set
Let and . Then , and the Lebesgue integral is the common value of and .
This is related to the definition given in the article as follows. Let . For each positive integer N, let denote the partition of the range of f given by . Let and denote the upper and lower Darboux sums for approximating the integral from the article. The supremum of for t in an interval is at most , and the infimum is at least , so that, by definition, we have and .
This proves that the definition given in the article is equivalent (in a fairly trivial "from the definition" way) to the Lebesgue approach. In other words, Lebesgue's definition of the integral really is trivially just given by the Riemann-Darboux integral of the distribution function . Hopefully this lays all further objections to rest. Sławomir Biały ( talk) 11:54, 21 June 2016 (UTC)
While this article is very long (and rightfully so), I believe it needs an "Applications" section. Otherwise the uninitiated reader will ask "What's the point?" To avoid having the applications section go unnoticed by readers who may not look far down for it, I'm going to follow the WP standard and put it early in the article since it should be as elementary as possible, and since our math articles start out simply before getting more complicated. Expansion of the section, while keeping it simple, would be welcome! Loraof ( talk) 20:11, 7 July 2017 (UTC)
Why can't I write about the usage of the integral in Kinematics or integrals of standard functions? There is no separate article for it. Should I create a new article? The reverse power rule and stuff like that is one the first things you learns in integral calculus. Lie Cleaner HK 17:32, 15 August 2017 (UTC)
The addition of the isn't needed here. This is a common abuse of notation that's generally understood from context. Moreover, the explanation is really getting out of scope for this article, especially for the lead since this the article is about integration, and not antidifferentiation. A footnote might be okay here, but that's still probably overkill. -- Deacon Vorbis ( talk) 16:53, 15 August 2017 (UTC)
Take a look here: /info/en/?search=Constant_of_integration#Necessity_of_the_constant Lie Cleaner HK 17:32, 15 August 2017 (UTC)
I apologize. I realize now that the +C is implied in the indefinite integral. I confused this with a more strict formula for the indefinite integral containing an initial condition. ScaAr ( talk) 21:30, 29 August 2017 (UTC)
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Much of it seems like an essay which is not good encyclopedic practice at least with the high standards of english wikipedia. — Preceding unsigned comment added by Yoandri Dominguez Garcia ( talk • contribs) 14:50, 15 June 2018 (UTC)
@ Rgdboer: You say "in-line references are not appropriate". Are you quoting the Wikipedia Manual of Style or Mathematics guidelines? I have looked in those documents and found nothing about in-line references being inappropriate in articles on the topic of mathematics, or less valuable than articles on other topics.
Over the years, nine Math articles have been raised to Featured Article status. Those nine, and their number of in-line citations are as follows:
There may be some truth in the idea that articles about Mathematics do not require as many in-line citations as articles on other topics. If that is so, the Mathematics guidelines should be amended to make it clear and put the matter beyond dispute. Until then, I think User:Yoandri Dominguez Garcia is justified in drawing our attention to the fact that this article does not meet Wikipedia’s standards regarding in-line citations. Dolphin ( t) 11:18, 30 October 2019 (UTC)
@ Deacon Vorbis: The reversion of my contribution concerning hyperbola quadrature and its place in the History of Integration suppresses useful information. "Unreferenced" says the edit summary, yet Introduction to the Analysis of the Infinite is part of the contribution. Although it is an Original Source there is an article on that book from 1748, and our article has several secondary sources, including Henk Bos who concurs with what was contributed here. Also, referring to History of logarithms there is a statement by D. T. Whiteside that supports the contribution. This material is very old so accusations of OR such as you raised July 18 this year in Talk:Natural logarithm are bogus. The time you put in to improve this Project is appreciated; however, it appears you are repressing information about the century (1647 to 1748), before natural logarithm, when hyperbola-quadrature was used. Why not tag the contribution with "citation needed" rather than revert. Restoration of the contribution is requested by this Talk, perhaps citing Whiteside. Please respond since dialogue here is required before appeal to a higher venue. — Rgdboer ( talk) 20:39, 30 October 2019 (UTC)
{{
math}}
/{{
mvar}}
, and some wasn't even italicized), but that can be fixed. There were also some problems with the style of the prose; that's more difficult to fix without having a source to refer to – another important reason to include them. For just one example, you said, "Promulgation of the hyperbola-quadrature by Huygens and Nicholas Mercator assured the transcendental function's acceptance."This is kind of confusing, and I, for one, wouldn't be able to clear it up without having a source to refer to. A claim such as this really needs a source anyway. – Deacon Vorbis ( carbon • videos) 21:02, 30 October 2019 (UTC)
I don't believe that "Signed Area" should redirect to this "Integral" page. When I hear "signed area" I don't think only of the "areas are negative below the x-axis" convention for integrals. I also think of the related-but-more-general concept of, say, considering areas of regions enclosed by counter-clockwise paths as positive and clockwise paths as negative. This comes up very naturally when considering, say, the Shoelace formula. -- Helopticor ( talk) 13:05, 19 April 2020 (UTC)
This is a small point. I spotted a mistake in the definition of the Riemann integral, which included the following segment:
This is a typical argument of the epsilon-delta type. The mesh of a partition is the width of the largest sub-interval formed by the partition. If the width of the largest sub-interval (with some index k which we don't need to know) is , this implies that for all sub-intervals are . No need to go at the level of indices or of taking into account the plurality in the notation: the notion of mesh does the job.
So the correct formulation should be (and using lower case delta makes the argument even clearer, showing that it is the familiar epsilon-delta argument):
I will try again, asking all those who want to revert my change to read the above comment and indicate where it goes wrong, if you find something wrong with it.
Dessources ( talk) 13:13, 15 August 2021 (UTC)
Finding area by integration on the area between curve y = f(x) and x-axis? — Preceding unsigned comment added by 129.232.97.252 ( talk) 16:56, 14 May 2022 (UTC)
[2] breaks out separate sections for analytical vs symbolic integration, but I was raised that analytical and symbolic mean the same thing in this context. Is there some different meaning I'm not aware of? Rolf H Nelson ( talk) 04:56, 4 January 2022 (UTC)
I also agree that the article is muddled: finding an antiderivative is described in both the "Analytical" and "Symbolic" subsections. The article can be improved.
I am trying to improve the lead sentence since it came up in the village pump as an example of something that needs work. My contribution is based on the suggestions from a WikiProject:Mathematics discussion Thenub314 ( talk) 16:19, 10 February 2023 (UTC)