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
