Volumes of Solids of Revolution. A Unified Approach

TL;DR

This paper introduces a unified double integral approach for calculating volumes of solids of revolution, simplifying classical methods and deriving Pappus' theorem as a special case.

Contribution

It presents a novel unified double integral method that encompasses classical disk and shell techniques and derives Pappus' theorem from this framework.

Findings

Unified double integral method for volume calculation

Classical methods recovered via Fubini's theorem

Derivation of Pappus' theorem from the new formula

Abstract

We present a method to compute the volume of a solid of revolution as a double integral in a very simple way. Then, we see that the classical methods (disks and shells) are recovered if this double integral is computed by each of the two possible applications of Fubini's theorem. As a further application we also show how Pappus' theorem is obtained from our formula.