Volumes by Cylindrical Shells

The shell method is simply another way to find the volume of a spun solid, by building it out of nested hollow tubes, like the layers of an onion or the rings of a tree. All we're really doing is slicing the region into thin strips that run parallel to the axis, and letting each one spin into a tube. The formula has three pieces: $r(x)$ (how far the strip is from the axis, the tube's radius), $h(x)$ (how tall the strip is, the tube's height) and $2\pi r(x)$ (the distance around the tube). Cut a tube open and flatten it, and it's almost a thin rectangular sheet: around times tall times thick. Add up all the sheets and you have the volume.

Why bother with a second method? Take a look at what the washer method would need for spinning $y = f(x)$ around the y-axis. The slices would be horizontal, so you'd have to write the radii in terms of $y$, which means solving $y = f(x)$ for $x$. For something like $y = 2x^2 - x^3$, that's miserable. Shells let you skip that.

When to use shells. Rotating $y = f(x)$ about the y-axis with washers would mean solving for $x$ in terms of $y$. With shells you keep $x$: a vertical strip at $x$ sweeps a shell of radius $x$ and height $f(x)$.

Radius and height:

• Radius = distance from the strip to the axis: $x$ for the y-axis, $2 - x$ for the line $x = 2$.
• Height = top minus bottom.

Disks and shells always give the same answer; choose whichever makes the integral easier.

A common mistake. It's tempting to use $x$ as the radius every time, because that's how the first examples go. That only works when the axis is the y-axis. The radius is the distance from the strip to whatever axis you're spinning around, so for the line $x = 2$ it's $2 - x$.

Shells model layered objects: tree rings, the walls of pipes and tanks, and the layers of an onion-skin pressure vessel.