Disc integration ($V=\pi {\int }_a^{b}r(x{)}^{2}dx$)

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.