Introduction

Move over Riemann and make room for Fermat! Most textbooks on the integral calculus focus heavily on the Riemann integral when introducing integration. This method is very effective in transitioning students from the finite (or macro) world of finding area geometrically to the infinite (or micro) world of finding area by integration. Once the notation and abstract idea of an area made up of an infinite number of infinitely thin slices is mastered, most textbooks move directly on to integration techniques. Finding areas using rectangles is usually not mentioned again except in review, to help students visualize a more difficult example, or when transitioning to finding volumes using double integrals.

Riemann used rectangles of uniform width. This is very handy when letting the width, dx, tend to zero. It also corresponds nicely to the definition of the derivative presented in most textbooks, in which the width h = xi+1 – xi in the denominator approaches 0. I still advocate introducing integration in this manner. However, there is no reason to stop there and move directly on to integration techniques.

Prior to Riemann, even prior to Newton and Leibniz, Fermat and others were finding areas using the sum of thin rectangles. However, Fermat's rectangles were not of uniform width. The width of Fermat's rectangles decreased based on a geometric series. Looking at Fermat's method directly after introducing the Riemann integral broadens the student's perspective on the integral calculus. Also, his techniques can be presented in an analysis course to provide depth to the material.