Finding Volume with the Matryoshka Doll Method (a.k.a. the Cylindrical Shell Method)

By Mark Ryan

If you’ve studied some integral calculus, you’ve no doubt run across the disk and washer methods for computing volume: cutting up a given solid into thin slices, each the shape of a circular disk or washer (a disk with a hole in its middle). You then add up (integrate) the volumes of all the slices to get the total volume.

As explained in this bonus chapter on the cylindrical shell method, you instead cut up the given solid into thin concentric cylinders and then add up the volumes of all the cylinders. The concentric cylinders fit inside each other, kinda like those nested Russian dolls. Can you picture how they fit inside each other?

Imagine a soup can that somehow has many paper labels, each one covering the one beneath it. Or picture one of those de-linter rolls with the sticky papers you peel off. Each soup can label or piece of sticky paper is a cylindrical shell — before you tear it off, of course. After you tear it off, it’s an ordinary rectangle. Because each unrolled cylinder becomes a rectangle, the formula for the area of a rectangle is the key to the cylindrical shell method.

Dig into more details about calculus with these downloadable bonus chapters.