# 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.

What’s the volume?

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:

Multiply the area by the thickness of the shell,

*dx*, to get its volume.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).