A Lost Theorem: Definite Integrals in Asymptotic Setting

TL;DR

This paper introduces an axiomatic theory of integration that simplifies the process of setting up definite integrals in calculus applications, avoiding Riemann sums and providing a more elegant proof of existence.

Contribution

It proposes a new axiomatic framework for integration that streamlines calculus applications and offers a historical perspective on the development of the theory.

Findings

Axiomatic approach simplifies integral setup in applications.

Proof of integral existence becomes more elegant.

Connects integration theory with calculus history.

Abstract

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.