Justifying Differential Derivations when Setting Up Definite Integrals.

This paper addresses the forming of integral expressions such as by differential or infinitesimal derivations. A rigorous justification of these handy informal derivations is a major pedagogical problem in calculus. In this paper, checks are presented by which one can usually verify–with clear geometrical and physical considerations–that integral expressions obtained by differential derivations are correct. Thereby students who are used to rigorous mathematics can begin to trust streamlined differential derivations that are commonly used in physics and engineering. One check is for integrals of one function, another similar check is for the case where the integrand is a product of two functions. It is also shown that, in these checks, so called second-order terms can be left out.