Volume of revolution via Cavalieri

Problem

Compute the volume of the solid of revolution, about the $x$ axis, of the region $0\le y\le f(x)$ with $a\le x\le b$, assuming that the cross-section at height $x$ is a disc of radius $f(x)$. (You are applying Cavalieri to the continuum.)

Solution

We slice the solid with planes perpendicular to the $x$ axis. At height $x$ the cross-section is a disc of radius $f(x)$, hence of area $A(x) = \pi\bigl[f(x)\bigr]^2.$ Cavalieri’s principle “in the continuum” states that the volume is the accumulation of the cross-sectional areas along $x$: $\boxed{V = \int_a^b A(x)\,dx = \pi\int_a^b\bigl[f(x)\bigr]^2\,dx.}$ This is exactly the formula of the disc method.

Links

Topics: Integral
Concepts: Disc method · Cavalieri’s principle · Volume of revolution
Methods: Cavalieri’s principle
Skills: Prove · Integrate
Exercise types: Proof