Volumes of Solids of Revolution / Method of Cylinders
In the previous section we started looking at finding volumes of solids of revolution.
In that section we took cross sections that were rings or disks, found the cross
sectional area and then used the following formulas to find the volume of the solid.
In the previous section we only used cross sections that were in the shape of a disk or
a ring. This however does not always need to be the case. We can use any shape for
the cross sections as long as it can be expanded or contracted to completely cover the
solid we’re looking at. This is a good thing because as our first example will show us
we can’t always use rings/disks.
Example 1
Determine the volume of the solid obtained by rotating the region bounded
by
and the
x
axis about the
y
axis.
Solution
As we did in the previous section, let’s first graph the bounded region and solid. Note that the
bounded region here is the shaded portion shown. The curve is extended out a little past this for
the purposes of illustrating what the curve looks like.
So, we’ve basically got something that’s roughly doughnut shaped. If we were to use rings on
this solid here is what a typical ring would look like.
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This leads to several problems. First, both the inner and outer radius are defined by the same
function. This, in itself, can be dealt with on occasion as we saw in a example in the
Area
Between Curves
section. However, this usually means more work than other methods so it’s
often not the best approach.
This leads to the second problem we got here. In order to use rings we would need to put this
function into the form
. That is NOT easy in general for a cubic
polynomial and in other cases may not even be possible to do. Even when it is possible to do this
the resulting equation is often significantly messier than the original which can also cause
problems.
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 Fall '08
 sc
 René Descartes, Euclidean geometry

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