Application of Integrals

Volumes by cylindrical Shells

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Here we consider the problem of finding the volume of the solid S obtained by rotating about the y -axis the region bounded by = and .

Here is an animation which shows how to generate the solid.

It is difficult to handle this problem by the method introduced previously. Let's take another look of the solid.

The animation above suggests that we may approximate the solid by cylindrical shells.

9 cylindrical shells :

18 cylindrical shells :

27 cylindrical shells :

As we can see from the graphs above that total volume of the cylindrical shells gets closer and closer to the volume of the solid as the cylindrical shells get thinner and thinner.

We divide the interval [] into n subintervals [] of equal width and let . If the rectangle with base [] and height rotated about the y -axis, then the result is a cylindrical shell with inner radius , outer radius and height as illustrated above. The volume of is

=  

=

                                                        =

 

Therefore, an approximation to the volume of S is given by

 

 

This approximation become better as . From the definition of an integral, we get

 

Thus the volume of S is .

 

In the same manner, if S is the solid obtained by rotating about the y-axis the region bounded by ( where ),  , , and , then we can get that the volume of S is

　

Let's conclude the methods of finding the volume of a solid of revolution with the following example.

To find the volume of the solid obtained by rotating about the y -axis the region between and .

Note that the intersection of the curves are ( ) and ( ).

 

A picture of the solid is shown below.

The region and a typical shell are shown above. We see that the shell has radius , and height . So the volume is

 

 

There is another way of looking at this.

The cross-section perpendicular to y -axis at a distance of y from the origin is an annular ring with inner radius and outer radius , so the cross-sectional area is

Therefore, the volume of the solid is

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