# Archimedes' Method for Computing Areas and Volumes - Exercise on Proposition 4 of The Method

Author(s):
Gabriela R. Sanchis

Investigate Archimedes' Proposition 4 below, which concerns a paraboloid (the solid obtained by rotating the curve $x=y^2$ about the $x$-axis) inscribed inside a cylinder (with radius $CD$ and height $AD$).

a) Describe the balancing relationship that is illustrated in the sketch.

b) Use this relationship to determine a relationship between the volume of the paraboloid and the volume of the cylinder.

c) Find a formula for the volume of the paraboloid in terms of $r$, the radius of the cylinder.