A Modern Spin on Archimedes’ Quadrature of the Parabola

Abstract

Summary In the 3rd century B.C.E., Archimedes wrote his treatise on the quadrature of the parabola, in which he laid out his solution to the problem of finding the area of a parabolic segment. Archimedes attacks the problem from two sides; first, he uses classical mechanics to derive an area formula, and then he uses pure geometry to prove his formula is correct. In this paper, we will present two modern approaches to this problem using calculus. The first approach can be presented to a calculus class that has learned about optimization methods and convergent geometric sequences, and the second can be presented after they have learned the Fundamental Theorem of Calculus.