5.3 · Volumes of revolution: disks and washers

Start with cross-sectional area

A slice perpendicular to a rotation axis is a disk if it has no hole, and a washer if it has a hole. Its area is $\displaystyle \pi R^2$ or $\displaystyle \pi(R^2-r^2)$, respectively. Multiply by slice thickness and integrate.

The squared difference is not the square of the difference: $\displaystyle R^2-r^2$ measures the area removed from the larger disk. Label both radii on a sketch before writing the integral.

$$V=\pi\int_a^b(R(x)^2-r(x)^2)\,dx$$

Measure each radius from the axis

When rotating around the x-axis, a nonnegative graph’s height can be its radius. When the axis is $\displaystyle y=k$, the radius is a vertical distance from $\displaystyle k$ to the boundary. The curve farther from the axis supplies the outer radius.

For the region between $\displaystyle y=x$ and $\displaystyle y=2$ on $\displaystyle [0,2]$ rotated about the x-axis, the outer radius is $\displaystyle 2$ and the inner radius is $\displaystyle x$. Rotating the same region about $\displaystyle y=2$ instead creates disks of radius $\displaystyle 2-x$.

Choose the variable from the perpendicular slice

For a vertical rotation axis, washers use horizontal slices and an integral in $\displaystyle y$. Solve the boundaries for $\displaystyle x$ as functions of $\displaystyle y$ when needed.

Volume units are cubic, since the area of a cross section has squared length units and thickness contributes one more. Negative volume is evidence of swapped radii or reversed bounds.