Volumes of Revolution

Solids of Revolution & Disks

A solid of revolution is formed by spinning a plane region around an axis. We find its volume by slicing it into thin pieces and integrating.

The disk method slices perpendicular to the axis of rotation. Each slice forms a disk of radius $r(x)$ and thickness $dx$.

$$V = \int_a^b \pi [r(x)]^2 \, dx$$

For a region with a hole, the washer method subtracts the inner hole using outer radius $R(x)$ and inner radius $r(x)$:

Common pitfall: Forgetting to square the radius in the disk method. A disk's area is $\pi r^2$, so the radius must be squared.

The Shell Method

The shell method slices parallel to the axis of rotation, cutting the solid into cylindrical shells. Unrolling a shell of radius $x$ and height $f(x)$ gives its volume.

$$V = \int_a^b 2\pi x f(x) \, dx$$

Disks and shells are two ways to slice the same solid. Choose whichever matches the geometry to keep your integral simple.

Common pitfall: Mixing up disk and shell setups, or dropping the $2\pi x$ factor in shells. Always sketch a typical slice first.