Cauchy's limit-sum definition of the Riemann integral for continuous functions is regarded to be central for the understanding of the two standard versions of the Fundamental Theorem of Calculus (FTC). In this article we take exception to this viewpoint and argue that Cauchy's definition is completely extraneous to the mathematical ideas and intuitions responsible for the initial development of integration during the second half of the seventeenth century. We describe modern approaches to teaching elementary integration that do not rely on Cauchy's definition of the integral, and are directly traceable to the mathematics of the second half of the seventeenth century. It is argued that, as one would expect from the historic development of the mathematical ideas related to elementary integration, these teaching approaches have a definite didactic advantage for students of the elementary integral. Finally, it is shown how our approach to integration leads naturally to important mathematical ideas such as Riemann and Darboux integration.
Anyone who has ever taught introductory calculus can attest to the fact that students seldom understand Riemann sums or the fact that the integral of a continuous function is a limit of a sequence of Riemann sums (see note 1.1). In a recent article, David Bressoud [5, p. 99] remarked about the Fundamental Theorem of Calculus (FTC):
There is a fundamental problem with this statement of this fundamental theorem: few students understand it. The common interpretation is that integration and differentiation are inverse processes. That is fine as far as it goes. The problem is that the definite integral has been defined as a limit of Riemann sums. For most students, the working definition of the definite integral is the difference of the values of “the” antiderivative. When this interpretation of the theorem is combined with the common definition of integration, this theorem ceases to have any meaning.
In this paper we argue that there is a direct cognitive evolutionary line for elementary integration that took form, mostly, in the second half of the seventeenth century, beginning with the work of Cavalieri (1635) and culminating with the work of Barrow (1635), Newton (1664), Gregory (1668), and Leibniz (1684), among others. This line of evolution has an associated teaching approach for integration, which we shall call, following Leonard Gillman , the axiomatic approach to the integral. There are various authors whose names have been associated with this teaching approach. Apparently, Stanislaw Saks gave the first theoretical exposition of this approach in 1933 in his monograph Theory of the Integral (published in English in 1937) , and Hans Hahn and Arthur Rosenthal developed it further in 1948 in their book Set Functions [24, p. 149; 38]. In 1968, Howard Levi applied the method to the elementary integral [33, p. 65] and Serge Lang gave an exposition of the integral by means of “area functions" (see note 1.2) [28, pp. 208-215]. Finally, in 1993, L. Gillman  presented the elementary integral following an axiomatic approach; he also presented a large collection of ingenuous heuristic principles useful in calculating the integrals that are usually found in calculus applications, such as area between curves, arc length, surface area, volumes of revolution, etc. It should be emphatically remarked that the axiomatic approach to the integral needs no mention of Riemann sums. Referring to his approach to the integral, Gillman [20, p. 17] remarked:
As a result, our intuition is relieved of the responsibility of making predictions about infinite processes. (We don't even mention Riemann sums.)
Gillman's statement markedly contrasts with that of Bressoud, and poses, in our view, an interesting problem regarding the lessons in the pedagogy of the elementary integral to be derived from the history of calculus. Of course, it goes without saying that modern analysis is practically unimaginable without Cauchy's limit-sum definition of the integral, and that his definition was instrumental in the eventual development of Darboux (see note 1.3) and Lebesgue integration. In light of the possibility of developing the elementary integral without invoking Riemann sums, the following question of a pedagogical nature gains paramount importance: How can Cauchy's limit-sum definition of the integral be considered as essential for understanding the FTC, especially when it followed the latter result by around a century and a half and, furthermore, played no role in any of the four (independent) proofs of the FTC developed by Barrow, Newton, Leibniz and Gregory during the second half of the seventeenth century?
In this work we argue that the axiomatic approach to the integral offers a definite cognitive advantage for the student of elementary integration, being totally consistent with the historical development of the calculus. Hans Freudenthal [18, 17] and others have made important remarks regarding the role that the history of mathematics can play in its teaching. The history of the evolution of mathematical ideas provides us with significant insights into the inner workings of the human mind as it tries to organize mathematical ideas into coherent bodies of knowledge. In this work we argue that the original insights and intuitions that gave rise to the notion of the integral and the discovery of the FTC are rather removed from Cauchy's conception of the area bounded by a curve as a limiting value of cumulative areas of approximating rectangles. We shall argue that the axiomatic approach to the elementary integral is the “historically correct” approach to teaching integration.
Notes for page 1:
1.1. The reference to “Riemann sums” in this connection is rather obscure and certainly a misnomer from a historical point of view. Bourbaki [3, p. 157] remarked that these sums could have more appropriately been named Archimedes or Eudoxus sums.
1.2. Actually, the phrase “area functions" is a misnomer. They should rather be called “additive functions,” since they can be used for all standard applications of the integral, not just area; for example, volume, surface area, arc length, etc.
1.3. However, as Gillman [20, Section 2, p. 17] appropriately remarked, Darboux integration is closely related to his axiomatic approach to the integral, being (as we shall see) related to the question of the existence of an integral in Gillman's sense.