How to Find the Volume of a Cylindrical Shape with the Nested-Russian-Dolls Method of Integration

Integration enables you to calculate the volumes of an endless variety of complicated shapes that you can’t handle with regular geometry. You can cut up a solid into thin concentric cylinders and then add up the volumes of all the cylinders. It’s kinda like how those nested Russian dolls fit inside each other. Or imagine a soup can that somehow has many paper labels, each one covering the one beneath it. Or picture one of those clothes de-linters 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.

Here’s a problem: A solid is generated by taking the area bounded by the x-axis,


and then revolving it about the y-axis. The figure below shows you the idea. The light gray area is the shape that’s been revolved about the y-axis to create the entire 3-D solid.

A shape sort of like the Roman coliseum and one of its representative shells.
A shape sort of like the Roman coliseum and one of its representative shells.

What’s the volume?

  1. Determine the area of a representative cylindrical shell.

    When picturing a representative shell, focus on a shell that’s in no place in particular. The figure shows such a generic shell. Its radius is unknown, x, and its height is the height of the curve at x, namely e to the power of x. If, instead, you use a special shell like the outer-most shell with a radius of 3, you’re more likely to make the mistake of thinking that a representative shell has some known radius like 3 or a known height like e to the power of 3. Both the radius and the height are unknown.

    Each representative shell, like the soup can label or the sticky sheet from a de-linter, is just a rectangle whose area is, of course, length times width.


    So now you’ve got the general formula for the area of a representative shell:

  2. Multiply the area by the thickness of the shell, dx, to get its volume.

  3. Add up the volumes of all the shells from 2 to 3 by integrating.


With cylindrical shells, it’s not always clear what the limits of integration should be. Here’s a tip. You integrate from the right edge of the smallest cylinder to the right edge of the biggest cylinder. And note that you never integrate from the left edge to the right edge of the biggest cylinder (like from –3 to 3).

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