Integration by Riemann triangles

Abstract

In this article we present a method of integration inspired by an example from Proofs Without Words II by Nelsen. Riemann rectangles under a curve are transformed into triangles. The limit of the collection of triangles gives us a new shape. The area under the original curve and the area of the new shape are the same. In this way, interesting connections between areas under different curves are established and integration formulae are obtained. Only elementary concepts like similarity and Riemann rectangles are used. A variety of examples is worked out to show the strength of the method. The method may be used for first year calculus students to deepen their understanding of the Riemann integral.