Talk:Integral

Analytical vs Symbolic?

[1] 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)

Proposed additions, sections shown, thanks

Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands.

The bracket integration method is a generalization of Ramanujan's master theorem that can be applied to a wide range of integrals.

• Gonzalez, Ivan; Jiu, Lin; Moll, Victor H. (1 January 2020), "An extension of the method of brackets. Part 2", Open Mathematics, 18 (1): 983–995, doi:10.1515/math-2020-0062, ISSN 2391-5455

• Rich, Albert; Scheibe, Patrick; Abbasi, Nasser (16 December 2018), "Rule-based integration: An extensive system of symbolic integration rules", Journal of Open Source Software, 3 (32): 1073, doi:10.21105/joss.01073

TMM53 (talk) 08:23, 2 January 2023 (UTC) TMM53 (talk) 08:23, 2 January 2023 (UTC)

I added this content and 2 references.TMM53 (talk) 03:12, 23 March 2023 (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)

Formal definition

Is there a reason this article doesn’t include the standard definition for the Riemann integral? i.e.

$$\int _{a}^{b}f(x)dx=\lim _{n\to \infty }\sum _{i=1}^{n}f(x_{i})\Delta x$$

211.30.47.108 (talk) 10:17, 7 November 2023 (UTC)

In your definition, what is $x_{i}$? There are several conventions for how it could relate to a, b, and n. This article presents one of these conventions, in the "Formal definition" section. I'm not an expert on the history, but I think that it's based on Riemann's original formulation. A different convention leads to the upper and lower Darboux integrals, which are a bit simpler.

So I think that you're asking why the article presents Riemann integrals instead of Darboux integrals. That's a fair question. I don't know the answer. Mgnbar (talk) 01:05, 8 November 2023 (UTC)

"Integration with other techniques" listed at Redirects for discussion

The redirect Integration with other techniques has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 January 31 § Integration with other techniques until a consensus is reached. Steel1943 (talk) 21:04, 31 January 2024 (UTC)

Who first “rigorously formalized” integration?

In the History section, the subsection Formalization begins with:

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann.

Even though what it means to “rigorously formalize” something is somewhat subjective, I would argue that Cauchy “rigorously formalized” integration (of piecewise continuous functions) some decades before Riemann. Indeed, the same reference (Katz 2009, pp. 776–777) seems to say the same thing:

Cauchy’s treatment of the derivative, although using his new definition of limits, was closely related to the treatments in the works of Euler and Lagrange. Cauchy’s treatment of the integral, on the other hand, broke entirely new ground. Recall that, in the eighteenth century, integration was defined simply as the inverse of differentiation. Even Lacroix wrote that “the integral calculus is the inverse of the differential calculus, its object being to ascend from the differential coefficients to the function from which they are derived.” Although Leibniz had developed his notation to remind one of the integral as an infinite sum of infinitesimal areas, the problems inherent in the use of infinities convinced eighteenth-century mathematicians to take the notion of the indefinite integral, or antiderivative, as their basic notion for the theory of integration. They of course recognized that one could evaluate areas not only by use of antiderivatives but also by various approximation techniques. But it was Cauchy who first took these techniques as fundamental and proceeded to construct a theory of definite integrals upon them.

In particular, it was Cauchy, not Riemann, who first used limits to define the integral of a function. Is there any reason not to change the text to reflect this?

LambdaP (talk) 14:39, 3 May 2024 (UTC)

Thanks for raising this issue. It would help to have a clearer statement of the timeline, because Cauchy and Riemann overlapped in time. According to Riemann integral (not a reliable source, I know), Riemann presented the Riemann integral in 1854. When did Cauchy do his integral work? It's not explicitly said at Augustin-Louis Cauchy or Cours d'Analyse.

Once we establish the basic facts, then it would be good to understand why so many authors seem to attribute the first rigorous integral to Riemann.

Once we understand that, if everything holds up, then multiple Wikipedia articles will need to be changed. Mgnbar (talk) 16:09, 3 May 2024 (UTC)

It is clear that Cauchy defined integrals as limits of sums of areas of small rectangles. But, I am not sure that he used a formal definition of limits. According to Cours d'Analyse, he used the informal (at that time) concept of infinitesimals. Moreover, having a rigorous formalization of integrals requires not only a formal definition of limits, but also the proof that the limit does not depend on the way of dividing the interval of integration. So, my interpretation of Katz's quotation is that "Cauchy was the first to define integrals from limits", but this does not imply that it is not Riemann who "first formalized rigorously integrals, using limits". So, unless better sources are provided, section § Formalization does not require to be changed. D.Lazard (talk) 16:42, 3 May 2024 (UTC)

Right. I went and read Cauchy’s Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal, which was published in 1823. I think the relevant part is in the vingt-unième leçon, starting on p. 81. On p. 83, the same leçon includes an explicit discussion that the way of cutting intervals does not change the limit value of the integral.

With respect to the infinitesimals, it's less clear, but the word doesn't seem to appear in the proof. LambdaP (talk) 20:54, 3 May 2024 (UTC)