Disk method

Volume of revolution as stacked circular disks: $V=\pi\int_a^b\left[r(x)\right]^2dx$ .

The disk method finds the volume of a solid formed by revolving a region about an axis, when every cross-section perpendicular to that axis is a solid circle (no hole). Each infinitesimally thin disk has radius $r(x)$ (the distance from the axis to the curve) and area $\pi[r(x)]^2$ , so $V=\pi\int_a^b\left[r(x)\right]^2dx$ .

Examples

Revolving $y=\sqrt{x}$ about the $x$ -axis on $[0,4]$ gives $V=\pi\int_0^4 x\,dx=8\pi$ .