= Disc integration
{title2=$V=\pi\int_a^b r(x)^2\,dx$}
= Disk method
{synonym}
For a <solid of revolution> formed by rotating $0\leq y\leq r(x)$ 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 $V=\pi\int_a^b r(x)^2dx$. If the rotated region instead has inner and outer radii $0\leq r(x)\leq R(x)$, 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.
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