Solid of revolution

A solid of revolution is the three-dimensional region swept out by rotating a planar region about an axis in its plane. Perpendicular slices are disks or annuli, making volume, center-of-mass, and moment-of-inertia integrals one-dimensional.

A meridian $r=r(\theta )$ has cylindrical radius $\rho =r\sin \theta$ and height $z=r\cos \theta$. On a branch with monotone height and filled horizontal disk sections, the volume between nearby horizontal planes is $dV=\pi {\rho }^2|dz|$, giving the formula. The conditions matter: a meridian that folds in height or bounds annular sections requires separate branches and inner-radius subtraction. A spherical angular wedge instead has volume $(2\pi /3)r(\theta {)}^3\sin \theta d\theta$ when the solid fills each radial segment. Both describe volumes, but they partition the solid differently.

For a solid of revolution formed by rotating $0\le y\le r(x)$ about the horizontal axis, with nonnegative continuous radius $r$ on $[a,b]$, each perpendicular slice is a disk of area $\pi r(x{)}^2$. Adding the slice volumes by a definite integral gives $V=\pi {\int }_a^{b}r(x{)}^{2}dx$. If the rotated region instead has inner and outer radii $0\le r(x)\le R(x)$, subtract the missing inner disks: $V=\pi {\int }_a^b[R(x{)}^2-r(x{)}^2]dx$. This is often called the washer method. A complete revolution is already included in the circular cross-sectional area, so no further factor of $2\pi$ belongs in the volume integral.