6.8 Volumes of Solids of Revolution

In this section we are asked to find the volume of a solid created when we revolve a section of the graph around a line, which we call the axis of revolution. This section of the graph can be the area beneath a curve or it can be the area between two curves. The axis of revolution can be the x-axis or the y-axis, or it could be another line, which would be designated in the problem.

Lets start with the easiest volume to find, using the area beneath a curve and the x-axis as our section of the graph and our axis of revolution, respectively.

Lets say we are given the curve

and asked for the volume of the solid created when we revolve the area below this curve from x=2 to x=9 around the x-axis.

The curve, when revolved around the x-axis, will give us a shape similar to a cylinder, and we can find the area of this cylinder by dividing it into tiny sections and adding their areas. The equation for the volume of our solid would look like this:

where r is equal to distance along the y-axis, or f(x), and h is equal to the distance along the x-axis, or dx. If we used infintely small sections to find the volume of our solid, our equation for volume would look something like this:

or:

So, if we plug in our values from the problem we get

and then we can solve for the volume:

so, the volume of the solid of revolution is

There are some other ways a question about solids of revolution can be asked.

For example, instead of revolving around the x-axis, you could be asked to revolve a shape around another line, like x=-2. In this case, you would end up with a hole through the center of your solid, so you have to subtract the volume of that hole. The equation for a problem like that looks like this:

or, more simply:

where R is the distance from the axis of revolution to the curve farthest away from it, and r is the distance from the closest curve to the axis of revolution. When you are only given one curve and asked to revolve it around a specified line, the second curve would be the x-axis. When you are given two curves, you have to determine which is closer to the axis of revolution and work from there.

Some helpful sites for more information:

http://chuwm2.tripod.com/revolution/volume.html

http://www.analyzemath.com/calculus/Integrals/volume_solid_1.html

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