Shell Method: Formula & Examples

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  • 0:04 The Shell Method Formula
  • 1:00 Understanding the Formula
  • 3:08 Using the Shell Method
  • 4:59 Lesson Summary
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Lesson Transcript
Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson you will learn how the shell method can be used to compute the volume of certain solids of revolution. This lesson requires familiarity with integrals.

The Shell Method Formula

Suppose you want to compute the volume of a solid of revolution, that is, a solid formed by sweeping a two-dimensional region around an axis, as you can see in the picture on your screen right now.

Solid of revolution formed by rotating a region R around the y-axis.
Solid of revolution

You may have already studied one method for finding the volume, the disk/washer method, in which the solid is sliced into thin disks or washers (disks with a hole in the middle). However, the volume of some solids of revolution is difficult or even impossible to calculate using the disk/washer method. Fortunately, we have another way to slice the problem (literally).

Any solid of revolution can be sliced into thin cylindrical shells that fit snugly within each other. Then each shell's volume can be computed and added up to get the volume of the whole solid. First, let's see what the formula looks like, which is:

Shell method formula

Now we'll take some time to understand how it's put together.

Understanding the Formula

Let's break down that formula we just looked at in the previous section of this lesson. The shell method relies on an easy geometrical formula. A very thin cylindrical shell can be approximated by a very thin rectangular solid. How? A shell is like the curved part of an aluminum can. If you cut the shell and unroll it so that it becomes flat, the result looks like a rectangle. If the cylindrical shell has a radius r and height h, then the unrolled shell is a rectangle of length 2 π r (that is, the circumference of the cylinder) and width h. But it also has a third dimension. As thin as the shell looks, it still has a nonzero thickness; we call it dr. After unrolling the shell, the result is, in fact, a rectangular prism whose height (thickness) is also dr. Thus, the volume of the shell is approximated by the volume of the prism, which is L x W x H = (2 π r) x h x dr = 2πrh dr.

One cylindrical shell shown in the solid.
Cylindrical shell

Finally, the shell method formula is obtained by adding up all of these shell volumes, and allowing dr to get infinitesimally small. The lower bound for integration (a) is the smallest x (or y) value at which the slicing begins, and the upper bound (b) is the largest x (or y) value at which the slicing ends.

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