Talk:Integral

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.

References

• 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)

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)

To specifically answer some of your points:

Definition of the integral

Cauchy seems to have been the first to define the integral of a function $f:[a,b]\to \mathbb {R}$ using the quantity $S(f,\sigma =(x_{0}=a,x_{1},\ldots ,x_{n}=b))=\sum _{i=0}^{n-1}f(x_{i})(x_{i+1}-x_{i})$, rather than “defining” the integral as an antiderivative. He was only interested in integrating functions with finitely many discontinuities, though, and in fact he mainly focuses on continuous functions. For such functions, he shows (implicitly using the fact that a continuous function $f:[a,b]\to \mathbb {R}$ is uniformly continuous), that a) the quantity $S(f,\sigma )$ converges to some limit value as the mesh of $\sigma$ tends to zero, and b) this limit value does not depend on the choice of partitions $\sigma$. He calls this limit value a definite integral, which he suggests we write $\int _{a}^{b}f(x)dx$, in passing saying this notation was “imagined by Mr. Fourier”.

To me, it is clear that Cauchy is the first to rigorously define integrals in the modern sense, unless somebody else did before him. He did it at the latest in 1823, thirty years before Riemann On the other hand, Cauchy seemed mainly interested in actually integrating functions, rather than studying which functions are integrable, or studying integrable functions as a class. As far as I'm aware, all he ever considers are piecewise continuous functions, which he shows to be integrable.

Riemann's contribution

I found a good StackExchange answer that discusses Riemann's contribution to the theory of integration. It is well written and has a lot of references and is well worth reading in full, but of particular interest for us is the following paragraph:

Riemann's nontrivial contributions to this topic were: (A) giving a necessary and sufficient condition for integrability based on the behavior of a function; (B) using this condition to prove the integrability of a certain function having a dense set of discontinuities; (C) putting the focus on the collection of functions that are integrable according to some notion of integrability, rather than defining a notion of integrability only for the purpose of being able to prove certain desired integrability properties. Regarding (B), I believe this was the first time a function that was continuous on a dense set and discontinuous on another dense set had been defined (or even contemplated, for that matter). A well known example of such a function is the ruler function, which is also called the Thomae function because it first appeared in an 1875 booklet by Thomae.

It's not quite clear what condition (A) is, but it seems to be some precursor to the modern theorem that a bounded function is Riemann-integrable iff it is continuous almost everywhere (see the StackExchange response for more details).

What I'm taking out of this is that we can probably say that Riemann can be credited with turning integrable functions into an object of study, and this is likely why so many people say that he's the first to rigorously define the integral. Incidentally, since Riemann's and Cauchy's definitions of the integral yield the same set of functions, we should maybe say that functions are Cauchy-integrable (or Cauchy-Riemann integrable) rather than Riemann-integrable, although that ship has sailed more than a hundred years ago.

LambdaP (talk) 15:43, 4 May 2024 (UTC)

References

This is all great to read. Thanks for putting it together.

But I worry that we're straying into Wikipedia:No original research, by analyzing these texts and coming to our own judgment based on our knowledge of the math. It would be safer if we had reliable secondary sources explicitly saying that so-and-so was the first to formalize integrals. Mgnbar (talk) 15:53, 4 May 2024 (UTC)

I agree. I personally think we have everything we need in (Katz 2009).

Chapter 22 is titled “Analysis in the Nineteenth Century”. In the chapter introduction (p. 765), we read:

In his calculus texts, Cauchy defined the integral as a limit of a sum rather than as an antiderivative, as had been common in the eighteenth century. His extension of this notion of the integral to the domain of complex numbers led him to begin the development of complex analysis by the 1820s. Riemann further developed and extended these ideas in the middle of the century.

Section 22.1 is “Rigor in Analysis”, and subsections 22.1.1-5 (“Limits“, “Continuity”, “Convergence”, “Derivatives”, and “Integrals”) are essentially all about Cauchy's work. The section opens with:

In spite of the appeal of Lagrange’s method in England, Cauchy, back in France, found that this method was lacking in “rigor.” Cauchy in fact was not satisfied with what he believed were unfounded manipulations of algebraic expressions, especially infinitely long ones. Equations involving these expressions were only true for certain values, those values for which the infinite series was convergent. In particular, Cauchy discovered that the Taylor series for the function $f(x)=e^{-x^{2}}+e^{-(1/x^{2})}$ does not converge to the function. Thus, because from 1813 he was teaching at the École Polytechnique, Cauchy began to rethink the basis of the calculus entirely. In 1821, at the urging of several of his colleagues, he published his Cours d’analyse de l’École Royale Polytechnique in which he introduced new methods into the foundations of the calculus. We will study Cauchy’s ideas on limits, continuity, convergence, derivatives, and integrals in the context of an analysis of this text as well as its sequel of 1823, Résumé des leçons données à l’École Royale Polytechnique sur le calcul infinitesimal, for it is these texts, used in Paris, that provided the model for calculus texts for the remainder of the century.

Section 22.1.5 opens with:

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.… 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.

Later:

In the second part of his Résumé, Cauchy presented the details of a rigorous definition of the integral using sums. Cauchy probably took his definition from the work on approximations of definite integrals by Euler and by Lacroix. But rather than consider this method a way of approximating an area, presumably understood intuitively to exist, Cauchy made the approximation into a definition.

Section 22.1.6 discusses Fourier's work, then in Section 22.1.7 (“The Riemann Integral”):

In 1853, Georg Bernhard Riemann (1826–1866) attempted to generalize Dirichlet’s result by first determining precisely which functions were integrable according to Cauchy’s definition of the integral $\int _{a}^{b}f(x)dx$.… Riemann now asked a question that Cauchy had not: In what cases is a function integrable and in what cases not? Cauchy himself had only shown that a certain class of functions was integrable, but had not tried to find all such functions. Riemann, on the other hand, formulated a necessary and sufficient condition for a finite function $f(x)$ to be integrable: “If, with the infinite decrease of all the quantities $\delta$, the total size $s$ of the intervals in which the variations of the function $f(x)$ are greater than a given quantity $\sigma$ always becomes infinitely small in the end, then the sum $S$ converges when all the $\delta$ become infinitely small” and conversely.

It seems pretty clear to me that Katz views Cauchy as having first rigorously formalized the integral, as part of a larger program of introducing rigor in analysis in general, and Riemann having expanded on Cauchy's work. LambdaP (talk) 17:42, 4 May 2024 (UTC)

It seems that Katz did a confusion between "definition" and "computation": If the above quotation would be taken literaly, this would mean that definite integrals, and the fundamental theorem of calculus would have been almost forgotten during the 18th century. On the other hand, it seems true that, during the 18th century, the standard method for computing an integral was to compute first the antiderivative. One must recall that Cauchy was not teaching to future academic people, but to future engineers. So, the fact that many antiderivatives of common functions cannot be written in closed form was certainly a strong motivation for emphasizing on integrals rather than on antiderivatives.

As Euler knew the concept of limits, it seems unbelievable that he did not know a definition of integrals in terms of limits. So, it seems difficult to decide who was the first to use limits for defining integration, and what is exactly the contribution of Cayley.

On the other hand, I disagree with formulations such as "the first to have formalized...": In this context, the concept of formalization as well as the concept of rigor has evolved over the time (let us recal that the first formal definition of the real numbers dates from the second half of the 19th century, and thus that Cayley did not have a formal definition of the real numbers used in its "formal" definition of integrals).

For these reasons, I suggest to theplace the sentence "Integration was first rigorously formalized, using limits, by Riemann" with Riemann was the first to provide a coherent theory of integration that includes as corollaries the fundamental theorem of calculus as well as other fundamental theorems such as Fubini's theorem and Stokes' theorem. D.Lazard (talk) 13:16, 5 May 2024 (UTC)

I presume you mean Cauchy rather than Cayley? :-)

I think what any of us finds believable or not should have less importance than what solid secondary sources are saying on the question. Katz is extremely clear that Cauchy is the first to have defined the integral as the limit of a sum (see above quotes). Here's a few other secondary sources saying the same thing.

Here's Lützen (p. 170):

Cauchy broke radically from his predecessors with his definition of the integral… Leibniz had considered integrals as sums of infinitesimals but from the Bernoullis onwards it had been customary to define integration as the inverse process of differentiation. This made the indefinite integral the primary concept and had made integral calculus an appendix to differential calculus. Fourier was the first to change this picture… he focused on the definite integral $\int _{a}^{b}f(x)dx$ (putting the limits of integration at the top and bottom of the integral sign is in fact Fourier's idea) and stressed that it meant the area between the curve and the axis (Fourier 1822, §229).

Cauchy followed Fourier when he focused on the definite integral, but instead of relying on a vague notion of area, Cauchy defined the definite integral as the limit of a "left sum". This was much more precise and it allowed him to prove that the integral exists for a continuous function.

further down (pp. 171-172):

Euler and his contemporaries had already used left sums to approximate integrals, and Lacroix and Poisson had tried to prove that they converge to the integral in a suitable sense. One can find many elements of Cauchy's arguments in these papers as well as in Lagrange's proof of the fundamental theorem of calculus (see (Grabiner 1981, Chapter 6)), and it is very possible that Cauchy built on these sources. Yet Cauchy's treatment is much clearer… [a]nd most importantly, Cauchy changed the technique from being a numerical approximation procedure to being a definition....

In the case of the integral the earlier approximation techniques led Cauchy to a definition which allowed him to prove the existence of the integral for a specific type of functions. No one seems to have asked this existence question before, nor could it have been answered with the earlier definition. Cauchy also proved general existence theorems in the theory of differential equations. Instead of asking how to integrate a special function or a special differential equation (that is, finding an analytic expression for the solution), Cauchy began the process of establishing the existence of the integral for a wide class of functions (or differential equations). He thus started an important process towards a qualitative mathematics which was carried further by Sturm-Liouville theory and by Poincare (see Chapter 11).

(Lützen, Jesper. "The foundation of analysis in the 19th century." A history of analysis (2003): 155-195.)

Here's Gray (p. 55):

In the century since Newton and Leibniz the fundamental theorem of the calculus had become regarded as allowing the integral of a function to be regarded as the opposite of its derivative; the integral of a function $f(x)$ is a function $F(x)$ with the property that ${\frac {d}{dx}}F(x)=f(x)$. As such, it is determined up to an additive constant.

Cauchy firmly reversed this trend, and restored an independent existence on the integral. In the second part of the Résumé of 1823 he defined the integral as a limit of sums of areas.

The Cauchy integral of a function of a real variable $y=f(x)$ that is continuous in a given interval was defined as follows. He divided the interval into n equal subintervals $[x_{i-1},x_{i}]$, and considered the sum $S=\sum _{i}(x_{i}-x_{i-1})f(x_{i-1})$. The integral will be the limit of this sum as $n$ tends to infinity.

(Gray, Jeremy. The real and the complex: a history of analysis in the 19th century. Cham: Springer, 2015.)

Besides, the first to provide a coherent theory of integration that includes the fundamental theorem of calculus as a corollary… is Cauchy, who proves the theorem in his Résumé (see e.g., Katz p. 778 or Lützen p. 171). Not sure about Fubini or Stokes' theorems, it's an interesting question for sure.

As an aside, shortly after he published his Résumé, Cauchy was scolded by Polytechnique for focusing too much on rigor and not on practical, engineering-oriented stuff. Quoting Katz again (p. 779):

There is a curious story connected with Cauchy’s treatment of differential equations. Cauchy never published an account of this second-year course, and it is only recently that proof sheets for the first thirteen lectures of the course have come to light. It is not clear why these notes stop at this point, but there is evidence that Cauchy was reproached by the directors of the school. He was told that, because the École Polytechnique [sic] was basically an engineering school, he should use class time to teach applications of differential equations rather than to deal with questions of rigor. Cauchy was forced to conform and announced that he would no longer give completely rigorous demonstrations. He evidently then felt that he could not publish his lectures on the material, because they did not reflect his own conception of how the subject should be handled.

I doubt practical matters of engineering was a strong driver of how he shaped his course prior to that point. LambdaP (talk) 21:25, 5 May 2024 (UTC)