Talk:Integral/Archive 5

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Terminology

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)

The point of the section "Terminology" is to define the terminology used in this article. Does this new version suit you? I have reinstated the bold characters, because it is customary on Wikipedia to highlight key definitions in bold characters. J.P. Martin-Flatin (talk) 11:21, 13 November 2015 (UTC)

I see that you have deleted the following subsection from the section "Terminology":

__________________________________________________________________________

Symbol dx

The symbol dx may have different interpretations depending on the theory of integration being used:

• If the underlying theory of integration is not important, dx can be seen as strictly a notation indicating that x is a dummy variable of integration.
• In Leibniz's notation, dx is interpreted as an infinitesimal change in x.
• If the integral is seen as a Riemann integral, dx indicates that the sum is over subintervals in the domain of x.
• In a Riemann–Stieltjes integral, dx indicates the weight applied to a subinterval in the sum.
• In Lebesgue integration and its extensions, dx is a measure, a type of function which assigns sizes to sets.
• In non-standard analysis, dx is an infinitesimal.
• In the theory of differentiable manifolds, dx is often a differential form, a quantity which assigns numbers to tangent vectors.

__________________________________________________________________________

Why did you do so?

I also see that you have added this sentence:

__________________________________________________________________________

Some authors place the symbol dx before the integrand, as in

$\displaystyle \int _{0}^{1}dx\,{\frac {3}{x^{2}+1}}.$

__________________________________________________________________________

I have never seen it used in practice. Do you have references? J.P. Martin-Flatin (talk) 11:32, 13 November 2015 (UTC)

1. No, WP:MOSBOLD does not say to bold key terms. The guideline is to bold the first appearance of the article title. 2. Yes, we introduce the notation as it's used in the article. This article concerns the integral over a real interval. There are other kinds of integrals discussed in the "Extensions" section, each of which has its own individual article, where notation can be introduced. These extensions are mentioned in a brief sentence in the lead, which is also appropriate per WP:LEAD. 3. This article is about the integral, not the many meanings of differentials in mathematics. We already have a separate article on that subject, and the content as written here was borderline WP:OR. It's better to keep the discussion as simple as possible. The likely audience of this article will just be confused by a long list of bullets about what "dx" means, especially if it appears before any sort of integral has actually been introduced. One needs context for these things. The section exists only to introduce terminology, not to engage in philosophical speculation about the meaning of that terminology. Let's try to write an article that might actually be useful for somebody. 4. I didn't add that sentence. But I have seen this in practice, not sure where. Sławomir
Biały 11:48, 13 November 2015 (UTC)

I have updated the heading of this section to clarify the scope of this discussion. To answer your points:

1. No, it is customary on Wikipedia to use bold characters to define key terms. But I will take out the bold characters since you dislike them.

2. The scope of this article is currently larger than what you claim. If we trim down section "Terminology and notation to adapt it to this reduced scope, then we need to delete a considerable amount of material further down, and most of section Extensions. This needs to be discussed and agreed upon. Imposing this reduced scope only to section "Terminology and notation makes no sense. So I am going to revert this change.

3. You missed the point. Moreover, the term "philosophical speculation" is judgmental. A short paragraph on the different meanings of dx (which I did not write by the way, just slightly edited) seems perfectly fine to me. Let us see what others think about it. In the meantime, I will leave it out.

4. This example was not there before your edit. If you did not add it, then it came by magic! I am going to take it out.

5. There is a term in the dictionary called compromise. You should look it up, you might find it instructive. It means that when two people disagree, each of them needs to make a step toward the other to solve the problem. J.P. Martin-Flatin (talk) 13:34, 13 November 2015 (UTC)

1. You're just wrong about this. There is no support for this in the manual of style. WP:MOSBOLD is very clear that boldface type should not be used in the article text, except in three very specific cases. "Defining key terms" is not on that list. If you believe that the manual of style should be changed, then by all means propose it at WT:MOS.

2. Top to bottom, the article is concerned almost exclusively with the one variable case. There is no need to reduce the scope of the article. It already is so reduced. The "Extensions" section exists to link to other notions of integration. These all have their own individual articles. I do agree that some of the content there could be trimmed down and summary style observed here.

3. I don't think you made a "point" for me to "miss". Whereas you apparently have either missed or dismissed my point. By the time the reader has reached the "terminology" section, he hasn't even been told what the integral is. It would never even occur to such a person to wonder what "dx" stands for. The entire notation $\int _{a}^{b}f(x)\,dx$ hasn't even been explained. Discussing that aspect of the notation only makes sense after the article has covered some meaningful content.

4. "This example was not there before your edit. If you did not add it, then it came by magic!" Has it occurred to you that someone else may have edited the article in the meantime?

5. Compromise is also achieved by discussion. So far you haven't addressed the central points of discussion. Nor have you made any new points.

While the current content, while it is not as bad as the original revision, still dwells too much on the multivariate case. Let's look at the current structure of the article:

Lead: The integral (in a real variable). History: History of the integral, real variable. Terminology: Integral sign, "dx", domain of integration. Then some stuff about integrals over surfaces, volumes, etc. Interpretations of the integral: One variable. Formal definition: One variable. Properties: One variable. Fundamental theorem of calculus: One variable. Extensions: Here we link to other articles, which discuss more advanced concepts of integration.

I don't mind a single sentence, as proposed by User:Ozob, that mentions integrals over other domains. But since the article does not discuss that case, I conclude that it is not right to cover that notation in the notation section that appears before the basic discussion of the integral. It is confusing to have notation there that isn't used in the article body. Sławomir
Biały 14:36, 13 November 2015 (UTC)

Finally, it's not clear why you want to move the mention of line integrals, surface integrals, and volume integrals out of the lead. The lead is supposed to summarize the article. This one sentence summarizes the "Extensions" section of the article, so it should stay there. Please justify in detail why you feel that this content is more appropriately covered as a merely notational issue in the "Notation and terminology" section. Also explain in detail why you believe on the one hand that the article is about multivariate integrals, and yet the lead should only cover the univariate case. Please also say why this is consistent with the following nutshell summary of WP:LEAD:

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.

Thanks, Sławomir
Biały 14:42, 13 November 2015 (UTC)

Sorry, I did not notice the modification by User:Ozob in the middle of your series of edits.

I am happy if you reduce the scope of the entire article to simple integrals over a real interval and trim down section "Extensions" drastically. This would help clarify the scope of this paper and its target audience, as mentioned in the next section. This would also address the issue of the multiple meanings of dx. I am not happy if you keep enforcing this policy only in section "Terminology and notation and not elsewhere, because that is inconsistent. J.P. Martin-Flatin (talk) 15:10, 13 November 2015 (UTC)

On the contrary, I've enforced this policy throughout the article. I have moved all of the content on differential forms, which was confusingly scattered throughout the article, into the "Extensions" section. It's pretty typical of mathematics articles about basic concepts to include generalizations sections like this. The focus of the article is still the basic case, though, and it's generally inappropriate for the article to take the more generalized perspective throughout. For example, the article group (mathematics) could everywhere be rewritten from the perspective that a group is a groupoid with only one object. But that would not really lead to a clear treatment of the subject. The situation is very similar here. I've reduced the length of two of the sections in "Extensions". The last part needs more careful work, and I don't have time to do it right now. Sławomir
Biały 15:49, 13 November 2015 (UTC)

In principle, I think the entire notation section could simply be eliminated. While at one point, the section served the purpose of introducing the integral, that content has since been absorbed into the History section. But it's now been totally repurposed to do something that would be much better done in context. The integral sign is now covered in the "History" section. Differentials are much better discussed in situ, because without the right context it's impossible to say what that notation actually means. Sławomir
Biały 15:04, 13 November 2015 (UTC)

I disagree. Defining the terminology and notation at the beginning of a long article is a best practice. J.P. Martin-Flatin (talk) 15:10, 13 November 2015 (UTC)

So is it appropriate to define terminology several sections before the concepts that the terminology refers to have actually been defined? I've never seen that in an article. But anyway, the current terminology section also introduces terminology that isn't used in most of the article. It's hard to see how that is a best practice. The mathematics good article Hilbert space, for example, does not begin by saying: "The inner product on a Hilbert space is sometimes written $\langle ,\rangle$, $\langle |\rangle$, $(,)$, or sometimes using a dot." That would not be a good way to begin the article, because we haven't even said what an inner product is. I'm astonished with the attitude that it would be appropriate to discuss notation before we can meaningfully attach a concept to the notation. That does not seem like a very good best practice. Sławomir
Biały 15:19, 13 November 2015 (UTC)

How about you delete most of the contents of section "Extensions", which would settle the issue? J.P. Martin-Flatin (talk) 15:35, 13 November 2015 (UTC)

I don't think "delete" is the right verb. Content should be selectively merged elsewhere. For instance, the current section on differential forms is in some places more detailed than the main article. Sławomir
Biały 15:39, 13 November 2015 (UTC)

Yes, I agree with you about the selective merge.

Your recent trimming of section Extensions is going in the right direction. I think we should go further. I have just transferred much material from subsection "Integrals of differential forms" to article Differential form. How about reducing "Line integrals" and Surface integrals down to one sentence each? J.P. Martin-Flatin (talk) 10:09, 14 November 2015 (UTC)

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)

Computation

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)

I think you'll find the scope of the article is already limited to one variable. Instead, it seems to me like you are the one proposing to generalize the subject of the article to be about all different kinds of integrals. That would require a major restructuring: effectively the entire article would need to be rewritten from scratch. I've given a list of what each section of the article covers, and it's clear that apart from the "Extensions" section, which exists mostly as a pointer to other notions of integration, the entire article, from the lead all the way down, exclusively concerns the one variable case.

I agree that both the computation section and the extensions section should be reduced in size. The emphasis in the lead on differential forms is not really appropriate either (WP:WEIGHT, WP:LEAD). I don't think a new article is needed. There are already articles on symbolic integration and numerical integration that can house this content. Sławomir
Biały 14:53, 13 November 2015 (UTC)

Yes, merging this material into the articles symbolic integration and numerical integration is also a possibility. What does User:Ozob think about it? J.P. Martin-Flatin (talk) 15:34, 13 November 2015 (UTC)

I haven't read all of this discussion in detail, and I don't understand the section naming on this talk page, but here are some comments:

• The focus of a Wikipedia article might not match the title of the article. This article seems focused on one-dimensional definite integrals. Let's improve it with that mission in mind. Later, if there is consensus that Wikipedia's Integral article should be a more general overview, merely linking to this article to handle a special case, then we can move articles to make that happen.
• Until that more general overview is in place (if ever), I agree that there should be a Generalizations section in this article, but that it should be shorter than it is currently.
• The article repeatedly blurs the distinction between integration and antidifferentiation. For example, the Symbolic subsection of the Computation section dwells on antiderivatives. Mgnbar (talk) 17:54, 13 November 2015 (UTC)

I view this article as an overview of integration, broadly construed. Historically, integrals in one variable came first, and they are still the most important case (the number of phenomena that can be modeled with a single variable is enormous). Because of that it is proper for this article to include a lot of discussion of the single variable case. However, it should not include every detail of the single variable case, and it should mention other generalizations: Stieltjes integrals and integrals with respect to general measures, multivariable integrals, integrals of differential forms, stochastic integrals, even integrals as a pairing between homology and cohomology. With that in mind I'd like to suggest the following changes:

• The content in the "Terminology and notation" section should be merged into the rest of the article, and the section itself should be removed. A proper discussion of notation depends on the reader knowing what is being notated, but the reader has not yet been introduced to any integrals. Integrating (ha ha) this section into the rest of the article will make the article more readable.
• The "Interpretations of the integral" section should have some discussion of contour and multivariate integrals as well as integrals in probability.
• There's too much detail on differential forms. Differential forms have their own article.
• There's also too much detail on numerical integration.
• The section on "important definite integrals" is so useless that I am going to remove it right now.

One last comment: Yes, people do use the notation $\int dx\,f(x)$. Yes, it hurts my eyes too. But it's in common use in physics and engineering (where you may even see $\int d^{3}x\,f(x,y,z)$ – oh, horror!). Ozob (talk) 00:11, 14 November 2015 (UTC)

Could you provide a reference (a textbook, not just lecture notes) using this bizarre notation? Then we could put it back in subsection "Variants" and mention explicitly that this notation is considered bad practice.J.P. Martin-Flatin (talk) 10:17, 14 November 2015 (UTC)

Any advanced physics textbook. It is common knowledge, so no reference needed for this. YohanN7 (talk) 10:21, 14 November 2015 (UTC)

By the way, while I also think it hurts the eye, the notation does make some sense under some circumstances where it is used. The integral may be a part of a larger expression, where f(x) plays the role of an operator (acting on what follows in the expression). If you'd like to include mention that this is "bad practice", then you'd need a reference for that. That it is ugly needs no mention or reference. It is obvious to the reader. YohanN7 (talk) 11:22, 14 November 2015 (UTC)

This introductory article primarily targets K-12 students in 12th form and undergraduates in first or second year. I am not sure they would be able to tell which notation is neat and which one is ugly.

Anyway, could we get back to the initial question: Should the section "Computation" be trimmed down drastically to finish refocusing the article on integrals over an interval of the real line? Thanks. J.P. Martin-Flatin (talk) 14:10, 14 November 2015 (UTC)

What is neat and what is ugly is highly POV. What could reasonable go in is where (predominantly mathematical physics) the particular notation is to be found.YohanN7 (talk) 10:53, 16 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)

I've cut the computation section a little. I still don't think the section is very good, but I don't know enough about numerical methods to really do a good job here. Ozob (talk) 22:54, 14 November 2015 (UTC)

In view of the lack of support for my proposal to cut down section "Computation" very significantly, I will not implement it. J.P. Martin-Flatin (talk) 15:18, 24 November 2015 (UTC)

Area under the curve

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)

We shouldn't do this with a hatnote, since that is a related use of the term (WP:RELATED). I'm inclined to think that a disambiguation page too fails the same test. This is the general article. We can easily refer to related, more specialized, articles here. But the use of the term "area under the curve" in pharmacokinetics is not different from the use of the term here. It is just more specialized. So I think that is best handled by a link to the more specific topic from this article on the general topic (see WP:DABCONCEPT). Sławomir
Biały 11:56, 15 December 2015 (UTC)

It seems to me that 'AUC' is what is used and would be looked up and there already is a disambigution page for that. Dmcq (talk) 10:33, 16 December 2015 (UTC)

Language redirection issue

"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)

Nationalist propaganda

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))

I've removed the unsourced content. Mindmatrix 21:31, 7 March 2016 (UTC)

Lebesgue integral diagram

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)

The picture shows partitioning the range, as you describe. The "area of a strip" is the measure of the width times the mesh of the partition. Indeed, the Lebesgue integral can be written as $\int f\,d\mu =\int _{0}^{\infty }\mu \{x\mid f(x)>t\}\,dt$, where the integrand is the area of an infinitesimal horizontal strip under the graph of the function. I don't see why "projection onto the x axis" is a helpful concept here. Does that clarify the nature of the Lebesgue integral in a way that the horizontal strips picture does not? Sławomir
Biały 12:22, 16 April 2016 (UTC)5 / 123

Each horizontal strip in the diagram represents a set of the form $\{(x,y):y_{1}<y<y_{2},y=f(x)\}$. One can certainly define the Lebesgue integral using sums of expressions like $(y_{2}-y_{1})\mu \{x:y_{1}<y<y_{2},y=f(x)\}$ which, if the function is continuous, is the area of one of the horizontal strips. But this is not what Lebesgue did. Instead, his integral is as you give: the sum of expressions like $y_{1}\mu \{x:y_{1}<y<y_{2},y=f(x)\}$ which in the diagram would, except for the middle region, correspond to two vertical strips. It is not easy to represent this diagrammatically. No diagram can really capture the essential difference which is using measure of sets of x values. Why? Because it is only when such sets are not finite unions of intervals that the Riemann and Lebesgue integrals differ.

Also, both Riemann and Lebesgue diagrams are slightly misleading as the function goes continuously to zero at the ends of the region of integration. This makes the horizontal subdivision idea (which can be made to work) not so simple. In the lower part of a diagram of such a function, the horizontal strips would not correspond to values of the function. The given prescription would need to be modified to incorporate these horizontal strips.

I suggest replacing the two diagrams with a more 'wavy' function that does not go to zero at the end points, and with, say, 4 local maxima. For the Lebesgue integral, draw horizontal lines dividing the range of the function and project their intersections of the curve to the x axis. Instead of colouring all the vertical strips this creates, choose one interval on the y axis and colour all 8 corresponding vertical strips to represent a typical term of the approximating sum.TerryM--re (talk) 22:44, 20 April 2016 (UTC)

"Each horizontal strip in the diagram represents a set of the form $\{(x,y):y_{1}<y<y_{2},y=f(x)\}$" No, this is not true. The horizontal strips are strips in the undergraph of f(x). It's not clear to me what $\{(x,y):y_{1}<y<y_{2},y=f(x)\}$ even means.

The diagram is correct. If you want a very explicit treatment of the Lebesgue integral, using a partition of the y-axis, see, for example, the proof of Theorem 1.17 in Rudin's Real and complex analysis. For more background, Williamson's "Lebesgue integration" gives at least three equivalent definitions of the integral; the one illustrated in our diagram and its accompanying text is discussed in section 3.5.

In fact, I'm still not clear what your exact objection is. The integrand I gave above, which you agreed with, was $\mu \{x|f(x)>t\}\,dt$. This is the area of the infinitesimal slab contained between two sublevel sets of f, just as illustrated in the diagram. Nothing about this requires that the function be continuous or "goes to zero". For example, we could draw a step function, and partition its undergraph according to this prescription. Indeed, we could do this for much more complicated functions too, but then our ability to illustrate things graphically is limited. But I would say that the diagram is correct, and illustrates precisely what is intended. It seems like you're reading into it requirements like continuity, which are inessential. You should study the diagram together with the text of the article, and the text of Lebesgue integral, to understand what it is supposed to be illustrating.

According to the Princeton Companion: "Lebesgue defined his integral by partitioning the range of a function and summing up sets of x-coordinates (or arguments) belonging to given y-coordinates (or ordinates)." Sławomir
Biały 00:11, 21 April 2016 (UTC)

This is really just hand waving. The essential difference between the Lebesgue integral, however one decides to divide the intervals, is that Lebesgue uses Lebesgue measure and Riemann uses Peano-Jordan measure. TerryM--re (talk) 03:21, 21 June 2016 (UTC)

That's certainly one important difference, but the measure alone does not tell you how to define the integral. Also, by "hand-waving", presumably you mean that it is text intended to convey an intuitive, rather than mathematically rigorous, idea. That is very similar to the content under discussion. Full details are given in the main article Lebesgue integral. Sławomir Biały (talk) 10:10, 21 June 2016 (UTC)

I have just discovered that discussion was on Wikipedia talk back in 2012 and Svebert made the same points as I do. I did agree that $\mu \{x|f(x)>t\}\,dt$ is equivalent to the Lebesgue integral. But it certainly doesn't represent a horizontal slab because ">" leaves it open ended, and it is not how Lebesgue defined it. It is also rather convoluted. (Actually it corresponds to integration by parts and can be applied equally well to the Riemann integral.) I certainly am not assuming continuity; the whole purpose of the Lebesgue integral was to deal with measurable functions in general. TerryM--re (talk) 03:21, 21 June 2016 (UTC)

" But it certainly doesn't represent a horizontal slab because ">" leaves it open ended" — Wrong. Try to draw this set. The reason this approach was settled upon for the article is not because it is necessarily one of among several ways that Lebesgue defined his integral, but because it is the most concise approach that still conveys an element of the intuition. (And I am not convinced that it, or a trivially equivalent approach, does not appear in the works of Lebesgue.) It is sourced to the book by Lieb and Loos. I think that is good enough. Finally, your objection explicitly concerned continuity: "No diagram can really capture the essential difference which is using measure of sets of x values. Why? Because it is only when such sets are not finite unions of intervals that the Riemann and Lebesgue integrals differ. Also, both Riemann and Lebesgue diagrams are slightly misleading as the function goes continuously to zero at the ends of the region of integration. This makes the horizontal subdivision idea (which can be made to work) not so simple. In the lower part of a diagram of such a function, the horizontal strips would not correspond to values of the function. The given prescription would need to be modified to incorporate these horizontal strips." I have already explained how the prescription deals with those horizontal strips. So, I assume from your latest reply that this objection has been satisfactorily resolved. What then remains? Sławomir Biały (talk) 10:33, 21 June 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 $f(E)\subset [0,b]$. Fix an $\epsilon >0$ and an integer N such that $m(E)/N<\epsilon$. For $r=1,2,\dots$, $r<Nb+1$, let $E_{r}\subset E$ denote the measurable set

$E_{r}=\{x\in E|(r-1)/N<f(x)\leq r/N\}.$

Let $S_{N}=\sum _{r}N^{-1}r\,\mu (E_{r})$ and $s_{N}=\sum _{r}N^{-1}(r-1)\,\mu (E_{r})$. Then $S-s<\epsilon$, and the Lebesgue integral is the common value of $\inf _{N}S_{N}$ and $\sup _{N}s_{N}$.

This is related to the definition given in the article as follows. Let $f^{*}(t)=\mu \{x|f(x)>t\}$. For each positive integer N, let $P_{N}$ denote the partition of the range of f given by $0<b/N<2b/N<\dots <(N-1)b/N<b$. Let $U(f^{*},P_{N})$ and $L(f^{*},P_{N})$ denote the upper and lower Darboux sums for approximating the integral $\int _{0}^{b}f^{*}(t)\,dt$ from the article. The supremum of $f^{*}(t)$ for t in an interval $(r-1)b/N<t<rb/N$ is at most $\sum _{k\geq r-1}\mu (E_{k})$, and the infimum is at least $\sum _{k\geq r}\mu (E_{k})$, so that, by definition, we have $S_{N}\leq U(f^{*},P_{N})$ and $L(f^{*},P_{N})\leq s_{N}$.

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 $f^{*}$. Hopefully this lays all further objections to rest. Sławomir Biały (talk) 11:54, 21 June 2016 (UTC)

Needs an applications section

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)

Integrals of standard function and usage in Physics

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)

You can write about integrals in kinematics, and you have. If it is to be kept, I have made a few small changes: [1]. For integrals of standard functions, see article Lists of integrals. - DVdm (talk) 07:56, 24 August 2017 (UTC)

addition of +C

The addition of the $+C$ 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)

I agree. The statement does not talk about the general anti-derivative and for theoretic purposes the constant of integration can be absorbed into the generic F(x). When one gets down to specific examples and problems it is important to make this constant explicit, but that is not what the article is about. I could support a footnote, but would not push for it. --Bill Cherowitzo (talk) 17:18, 15 August 2017 (UTC)

Not only is it not needed. It would make the statement even wrong. - DVdm (talk) 17:22, 15 August 2017 (UTC)

The problem is that you are confusing readers by making them think that the antiderivative represented by an integral only has the unique result of C=0. This also implies that integration magically gives you the initial conditions. This is wrong. ScaAr (talk) 17:22, 15 August 2017 (UTC)

The addition of C is not wrong, it perfects the statement. If you think otherwise then you do not undertstand what you are talking about. ScaAr (talk) 17:26, 15 August 2017 (UTC)

I think you are confused. We write F(x) = int( x^2 dx ) ==> F(x) = 2x + C. - DVdm (talk) 17:30, 15 August 2017 (UTC)

I think "the antiderivative" is not a very common idiom, and the lead currently could be read as failing to distinguish between specific antiderivatives and the indefinite integral. This should be clarified if possible, without sacrificing readability. Sławomir Biały (talk) 17:32, 15 August 2017 (UTC)

I'd certainly be okay with something like "... a function F (which, if it exists, is not unique) whose derivative ...". Not sure if that's maybe a bit too awkward though. --Deacon Vorbis (talk) 17:37, 15 August 2017 (UTC)

Take a look here: https://en.wikipedia.org/wiki/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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This article is devoid of sources. Sad.

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)

Currently the article has more than 20 titles in a bibliography and more than 10 links to on-line texts. Furthermore, the template for integrals at the end of the article links to many types of integral where specialized sources can be found. As an article about a common college topic found in textbooks, in-line references are not appropriate. — Rgdboer (talk) 01:37, 30 October 2019 (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:

• Pi 223 in-line Citations

• Euclidean algorithm 158 in-line Citations

• Logarithm 109 in-line Citations

• Group (mathematics) 79 in-line Citations

• Parity of zero 75 in-line Citations

• 0.999… 71 in-line Notes and Citations

• Problem of Apollonius 66 in-line Citations

• Polar coordinate system 22 in-line Citations

• 1-2+3-4+… 21 in-line Notes and Citations

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)

Hyperbola quadrature

@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)

@Rgdboer: Please see WP:BURDEN. You added quite a lot of information without providing any sources. Providing your own analysis of primary sources (if that's what you did; I can't tell) is most definitely OR. Less importantly, the material added wasn't well-formatted (e.g. there were a lot of minor grammar problems, none of the math was placed inside {{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)

Signed Area redirect?

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)

Formal definition of the Riemann integral

This is a small point. I spotted a mistake in the definition of the Riemann integral, which included the following segment:

For all $\varepsilon >0$ there exists $\Delta _{i}>0$ such that, for any tagged partition $[a,b]$ with mesh less than $\Delta _{i}$,

$\left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .$

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 $<\delta$, this implies that for all sub-intervals $\Delta _{i}$ are $<\delta$. 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):

For all $\varepsilon >0$ there exists $\delta >0$ such that, for any tagged partition $[a,b]$ with mesh less than $\delta$,

$\left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .$

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)

How can something like "There exists delta such that <expression involving Delta_i>" be possible correct? I have reverted again. - DVdm (talk) 14:05, 15 August 2021 (UTC)

Could you please be more precise in the formulation of your objection. What expression are you referring to?

If you mean the following expression:

$\left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon$ (*)

please note that this expression does not contain $i$ as a free variable. Here $i$ is an index that runs from 1 to $n$, and expression

$\left|S-\sum _{j=k}^{n}f(t_{k})\,\Delta _{k}\right|<\varepsilon$

is strictly equivalent to it.

Dessources (talk) 14:59, 15 August 2021 (UTC)

I see what you mean. The $\Delta _{i}$ are defined in the previous paragraph. Let me propose a version that may accommodate your concern before reverting my change.Dessources (talk) 15:07, 15 August 2021 (UTC)

I did not find a way to make things clearer without repeating what is already said. When we say (*) is satisfied for any tagged partition, we refer to the definition just given of a tagged partition. Any tagged partition is defined by the finite increasing sequence of $x_{i}$'s, from $a$ to $b$, described in the first paragraph, which mechanically imply the width of the sub-intervals, Δ_i = x_i−x_i−1, and the $t_{i}$'s that fall within these intervals. When we take any tagged partition, we automatically get the $x_{i}$'s and $t_{i}$'s that define it, and thus also the $\Delta _{i}$'s.

Finally the error I corrected is obvious when one observes that it makes no sense to refer outside an expression to a variable which is bound in the expression, as was the case with the index $i$ that I removed. This alone is sufficient to justify the correction.

Dessources (talk) 15:52, 15 August 2021 (UTC)

@Dessources: Ah yes, I fumbled with the verification of the cited source. I overlooked the actual definition in section 8.5. I only had a look at the top of the page. My bad! - DVdm (talk) 07:56, 16 August 2021 (UTC)

Integration - calculas

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)

Analytical vs Symbolic?

[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 agree. It's not clear what the distinction is supposed to be. In fact, there is a great deal of overlap in the content, as it is currently written. Unless somebody chimes in with a strong explanation, I'd support merging the two sections. Mgnbar (talk) 13:40, 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.

• The main division is usually between those methods that find a formula containing well-known functions, and those methods that directly find a numerical value. I have seen the latter methods referred to as "approximate integration". But Wikipedia already has articles on symbolic integration and numerical integration so they are probably the best terms to use.

• I have seen methods of solving differential equations classified as "graphical", "numerical" or "analytical". But I am not sure how much the term analytical integration is used. There could be a distinction between methods that use clever mathematical analysis thinking and those that use brute-force calculation. Or perhaps analytical integration only applies to analytic functions. The term symbolic integration is probably becoming more popular because it is used in computer algebra systems.

• The current "Analytical" and "Symbolic" subsections both mention methods that find a symbolic representation as an infinite series which is then evaluated numerically. I have seen such methods classified as "approximate", and they could go in the "Numerical" subsection. But it might be better to have them in a separate subsection under the traditional name "Integration by series".

• I do not agree with the "Analytical" subsection where it says that "The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus". A numerical method such as counting squares under a graph is much simpler to explain.

• Also, I would change the name of the "Mechanical" subsection to "Analogue" and include links to both differential analyser and electronic analogue computer. JonH (talk) 16:43, 16 May 2022 (UTC)

Lead Sentence

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)