Volumes by Disks and Washers

The disk and washer methods are simply a way to find the volume of a solid made by spinning a flat region around a line, like a vase on a potter's wheel. All we're really doing is cutting the solid into thin coins and adding up their volumes. The formula has three pieces: $R(x)$ (the outer radius, from the axis out to the far edge), $r(x)$ (the inner radius, out to the near edge, zero if there's no hole) and $\pi\left[R^2 - r^2\right]$ (the area of the big circle minus the hole). Multiply each slice's area by its thickness $dx$, add them up, and you have the volume.

It's the same slice-and-add idea as the area problem, one dimension up. For area we added thin rectangles, height times $dx$. Now each slice is a thin coin, and its volume is its face area times $dx$. A coin's face is a circle, so the area is $\pi R^2$, and if the region doesn't touch the axis, the coin has a hole and becomes a washer.

A thin slice of the solid, perpendicular to the axis, is a disk of radius $R$ (area $\pi R^2$) or a washer with outer radius $R$ and hole radius $r$ (area $\pi R^2 - \pi r^2$). Multiply by the thickness $dx$ and integrate.

Radii are distances to the axis. Rotating around $y = 1$ instead of the x-axis, a point at height $y$ is $1 - y$ from the axis (or $y - 1$ if it's above).

A common mistake. It's tempting to subtract the radii first and then square, $\pi(R - r)^2$, because $R - r$ looks like the washer's width. It isn't the area of a washer: the washer is a big circle with a small circle removed, so its area is $\pi R^2 - \pi r^2$. Square each radius separately.

Around the y-axis, slice horizontally and write the radii as functions of $y$.

Lathes and 3D printers build parts exactly like this, and engineers compute the volume (and so the mass and cost) of turned parts such as shafts, bottles and nozzles with the washer method.