# Talk:Integral

Integral was a good articles nominee, but did not meet the good article criteria at the time. There are suggestions below for improving the article. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
 October 23, 2006 Good article nominee Not listed
To-do list for Integral:
 Here are some tasks awaiting attention: Article requests : * the article seems to lack focus and order, and there is no table of contents. Also, brief discussions on the general properties of the integral such as being a linear functional, along with two brief sections on the two definitions of the integral. treatment of integrals with regard to differential forms. Some images to illustrate the Informal discussion section. Like what?--Cronholm144 21:49, 28 June 2007 (UTC) A (sub)section on "Properties of integrals" covering general properties as a linear functional, Fundamental theorem of Calculus, etc. Copyedit : * Once major changes are complete, a thorough copyedit for flow and consistency is in order. Expand : * the section on Computing integrals could do with some expansion.

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## Computation

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. 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 ${\displaystyle \int dx\,f(x)}$. Yes, it hurts my eyes too. But it's in common use in physics and engineering (where you may even see ${\displaystyle \int d^{3}x\,f(x,y,z)}$ – 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). 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). 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 ${\displaystyle \int f\,d\mu =\int _{0}^{\infty }\mu \{x\mid f(x)>t\}\,dt}$, 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? 12:22, 16 April 2016 (UTC)5 / 123
Each horizontal strip in the diagram represents a set of the form ${\displaystyle \{(x,y):y_{1}. One can certainly define the Lebesgue integral using sums of expressions like ${\displaystyle (y_{2}-y_{1})\mu \{x:y_{1} 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 ${\displaystyle y_{1}\mu \{x:y_{1} 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 ${\displaystyle \{(x,y):y_{1}" No, this is not true. The horizontal strips are strips in the undergraph of f(x). It's not clear to me what ${\displaystyle \{(x,y):y_{1} 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 ${\displaystyle \mu \{x|f(x)>t\}\,dt}$. 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)." 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 ${\displaystyle \mu \{x|f(x)>t\}\,dt}$ 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 ${\displaystyle f(E)\subset [0,b]}$. Fix an ${\displaystyle \epsilon >0}$ and an integer N such that ${\displaystyle m(E)/N<\epsilon }$. For ${\displaystyle r=1,2,\dots }$, ${\displaystyle r, let ${\displaystyle E_{r}\subset E}$ denote the measurable set

${\displaystyle E_{r}=\{x\in E|(r-1)/N

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

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

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