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My sketchy understanding of the (no doubt long) history of integration theory is that the first integration theory was created by Riemann as part of his work on trigonometric series ("Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe", according to Wikipedia).

Other integration theories followed. The standard one today is the Lebesgue integral, which is used, among other places, in the theory of probability.

The Riemann integral is not as useful as the Lebesgue integral, but is still a "modern" integration theory, and is comparable to the Lebesgue integral.

My question is whether this was really the first "modern" integration theory. Did Riemann come up with it out of the blue? Were there precursors? And if it was his original creation, how much is known about how he developed the notion?

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    $\begingroup$ What precisely do you mean by an "integration theory"? Was Archimedes's method of exhaustion an integration theory? What about Fermat's computation of quadratures? Or the (inverse) relation between computing tangents and computing areas? If what you are after is a "general" theory of definite integrals, the definition via Riemann sums has a clear precedent in Cauchy. $\endgroup$
    – Andrés E. Caicedo
    Commented Feb 21, 2015 at 6:24
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    $\begingroup$ A reference you may find useful: History of measure theory by Djura Paunic, in the Handbook of measure theory, vol 1. $\endgroup$
    – Andrés E. Caicedo
    Commented Feb 21, 2015 at 6:25
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    $\begingroup$ Another text you should consult is Thomas Hawkins's Lebesgue's theory of integration. Its origins and development. $\endgroup$
    – Andrés E. Caicedo
    Commented Feb 21, 2015 at 7:00
  • $\begingroup$ @AndresCaicedo thank you for the book references. Do you have a good online source for Cauchy's theory. I'm in a place where it is not so easy to get hold of books. $\endgroup$
    – Faheem Mitha
    Commented Feb 21, 2015 at 15:42
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    $\begingroup$ If you Google "Cauchy definition of integral" you should get several results (some papers in repositories like JSTOR, or in their authors' webpages, some questions on Math.Stackexchange, etc). $\endgroup$
    – Andrés E. Caicedo
    Commented Feb 21, 2015 at 15:53

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The first rigorous integration theory in due to Eudoxus and Archimedes. It is called the method of exhaustion, and it allowed Archimedes to find the volumes of the balls, pyramids, cones, areas of segments of parabolas etc. (By the way, it was proved in 20th century that one cannot find the volume of a pyramid by pure geometric methods, some form of integral is indeed necessary here).

In the modern era, the first rigorous definition of the definite integral is due to Cauchy. Cauchy's definition of the integral is close to that of Riemann, the main difference is that he takes equidistant points in the partition to define the integral sums, and assigns specific points where the values of the function are taken. This gives the same result for continuous functions as Riemann's integral. Cauchy's approach is used sometimes in elementary calculus courses, because it is conceptually simpler than the Riemann approach.

It is not true that Riemann integral is "not as useful" as Lebesgue's integral. Because in many cases the functions we integrate are continuous. For example in all books on Complex Analysis, Riemann integral is used. It is also taught in almost all courses of real analysis, with very few exceptions, before the Lebesgue's integral, even in the Bourbaki course.

Sources. Medvedev, Development of the concept of integral, Moscow 1974. I. N. Pesin, Development of the concept of integral, Moscow 1966 (There is an English translation). N. Bourbaki, Elements of the history of mathematics. Translated from the 1984 French original by John Meldrum. Springer-Verlag, Berlin, 1994. N. Bourbaki, Functions of a real variable, Elementary theory. Translated from the French, Springer 2004.

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  • $\begingroup$ Thank you for the answer. Do you have a good online reference for the Cauchy theory? I don't remember it with any clarity, but had the impression that it is a much more restricted theory than the Riemann integral. I actually recall it as being some version of an antiderivative, which may be false. $\endgroup$
    – Faheem Mitha
    Commented Feb 21, 2015 at 15:46
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    $\begingroup$ jstor.org/stable/2007121?seq=1#page_scan_tab_contents $\endgroup$
    – Alexandre Eremenko
    Commented Feb 21, 2015 at 18:40
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    $\begingroup$ digitalcommons.ursinus.edu/cgi/… $\endgroup$
    – Alexandre Eremenko
    Commented Jan 16, 2021 at 3:41
  • $\begingroup$ What is the name (or the author name) of the theorem on the non-geometric integrability of pyramids? $\endgroup$
    – Mauricio
    Commented Sep 10, 2021 at 14:35
  • $\begingroup$ The name is Max Wilhelm Dehn. He was Hilbert's student when he proved this (and solved one of the Hilbert problems from the famous list). I suppose this was his first work. Later he became famous for "Dehn twists" and other 2-d geometry/topology. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 10, 2021 at 17:15
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I seem to remember a description of the origin of the Riemann integral that goes like this.

Integrals were widely used ever since Newton and Leibniz. But everyone just assumed all functions (or at least bounded functions on a finite interval) had integrals. In due time, the abstract notion of "function" came in. Then it was not so clear that all functions have integrals. At one point Cauchy published an argument that continuous functions have integrals. (Looking back today, we can see the notion of uniform continuity in the proof.) When Riemann looked at the proof he said: Here is what's going on: if we take such-and-such as the definition for "integral", then Cauchy showed that continuous functions satisfy it. But look, some discontinuous functions also satisfy it. So Riemann didn't just invent the definition from scratch. He distilled it from a proof of Cauchy.

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  • $\begingroup$ That's interesting. Does this come from Riemann's correspondence? $\endgroup$
    – Faheem Mitha
    Commented Feb 21, 2015 at 22:57
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    $\begingroup$ Fourier in 1822 gave first definition of a function that was not tied to analytic expressions: "In general, the function f(x) represents a succession of values or ordinates each of which is arbitrary... We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever..." In practice however he still assumed that all functions are continuous. Cauchy's version was even more traditional. Riemann might have been influenced by Dirichlet who accepted Fourier's definition and implications. www-history.mcs.st-and.ac.uk/HistTopics/Functions.html $\endgroup$
    – Conifold
    Commented Feb 23, 2015 at 0:56

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