Some volumes of revolution are more easily computed with cylindrical shells.

More than one method

In this section, we will show you a new method for computing volumes of solids of revolution. Consider the following region bounded by and :

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If this region is rotated around the -axis, it is possible, but inconvenient, to compute the volume of the resulting solid by the methods we have used so far. The issue is that there are two “kinds” of cylindrical cross-sections:
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As we see above, some of the cylindrical cross sections are defined by the line that goes from to , and others are defined by the line that touches at both ends. To compute the volume using accumulated cross-sections, we need to break the problem into two integrals:
  • an integral that computes the volume of the region bounded by and the line , rotated about the -axis, and
  • an integral that computes the volume of the region bounded by , and the line , rotated about the -axis.

Since we are rotating around the -axis, we should look at and .

With this in mind, we can compute our volume with: While we have successfully solved this problem, it wasn’t easy. Let’s see another, perhaps easier method to solve this problem. If instead we consider a vertical rectangle of height (just like we did when we computed areas of regions between curves!) and width , and we additionally rotate this rectangle around the -axis, we get a thin “shell” or hollow-tube:
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Here the infinitesimal change in volume is:
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Integrating will give us our desired volume.

Shells around the axes

Let’s start by actually doing our motivating example above.

Comparing our work above to our earlier work, we see that using shells not only solves the problem with just one integral, we see that the integral itself is somewhat easier than those in the previous calculation! Things are not always so neat, but it is often the case that one of the two methods will be simpler than the other, so it is worth considering both methods.

Shells around other lines

What if we had wanted to rotate the region from the last example about the line instead of the -axis? In this case, we draw a picture and work much the same way as before. Let’s see an example.