The disk, washer, and cylindrical shell methods all compute volumes of solids of revolution. The best method is usually the one that describes the region with the fewest pieces and the simplest boundary functions.
1. Disk method
A disk is a washer with inner radius zero. If a region under \(y=R(x)\) is rotated about the x-axis,
\[V=\pi\int_a^b R(x)^2\,dx.\]
Example: rotate the region under \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) about the x-axis:
\[V=\pi\int_0^4(\sqrt{x})^2\,dx=\pi\int_0^4x\,dx=8\pi.\]
2. Washer method
A washer has outer radius \(R\) and inner radius \(r\):
\[V=\pi\int_a^b\left(R(x)^2-r(x)^2\right)\,dx.\]
Use washers when slices perpendicular to the axis of rotation produce rings. Distances must be measured from the axis, not simply copied from y-values when the axis is shifted.
For example, rotating the region between \(y=2\) and \(y=x\) on \([0,2]\) about the x-axis gives
\[V=\pi\int_0^2(2^2-x^2)\,dx=\frac{16\pi}{3}.\]
3. Cylindrical shell method
A vertical strip rotated about the y-axis forms a cylindrical shell. Its volume element is circumference times height times thickness:
\[V=2\pi\int_a^b(\text{radius})(\text{height})\,dx.\]
For rotation about the y-axis, the radius is often x. For rotation about a shifted vertical line \(x=c\), the radius is a distance such as \(|x-c|\).
4. Compare washers and shells on the same region
Take the region between \(y=2x\) and \(y=x^2\) for \(0\le x\le2\), rotated about the y-axis.
Shells: radius \(x\), height \(2x-x^2\):
\[V=2\pi\int_0^2x(2x-x^2)\,dx=\frac{8\pi}{3}.\]
Washers: solve for x in terms of y. The right boundary is \(x=\sqrt{y}\) and the left boundary is \(x=y/2\), with \(0\le y\le4\):
\[V=\pi\int_0^4\left[(\sqrt{y})^2-\left(\frac y2\right)^2\right]dy=\frac{8\pi}{3}.\]
Both are correct, but the shell setup avoids solving both curves for x and is shorter.
5. A practical decision rule
| Question | Likely method |
|---|---|
| Do perpendicular slices make solid circles? | Disks |
| Do perpendicular slices make rings with a hole? | Washers |
| Do parallel slices make cylinders with simple height? | Shells |
| Would one method require solving for the other variable? | Try the other method |
| Does the region change boundary partway through? | Choose the method with fewer split integrals |
6. Shifted axes
When rotating around \(y=k\), radii are vertical distances such as \(|f(x)-k|\). Around \(x=c\), shell radii are horizontal distances such as \(|x-c|\). Drawing the axis explicitly prevents sign and radius errors.
7. Visualize before integrating
Use the Solid of Revolution visualizer to display the generating region, axis, outer and inner surfaces, and representative cross-sections. The visualization does not replace the calculus setup, but it makes incorrect radii or an unintended axis much easier to spot.
Common mistakes
- Using function values as radii when the axis is shifted.
- Reversing outer and inner radii.
- Forgetting the factor \(2\pi\) in the shell method.
- Using shell height as “top plus bottom” instead of top minus bottom.
- Mixing x-bounds with a dy integral or y-bounds with a dx integral.
- Choosing a method before sketching the region.
Review area between curves before setting up a rotated region, and compare with the introductory solid-of-revolution guide.