A Lost Theorem: Definite Integrals in an Asymptotic Setting

We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant than the conventional one. We also discuss an interesting connection between our approach and the history of calculus. The article is written for readers who teach calculus and its applications. It might be accessible to students under a teacher’s supervision and suitable for senior projects on calculus, real analysis, or history of mathematics. Here is a summary of our approach. Let ρ : [a, b] → R be a continuous function and let I : [a, b] × [a, b] → R be the corresponding integral function, defined by y I(x, y) = ρ(t)dt. x