Calculus
Name______________________
Volumes of Revolution: Making a Wise Decision When Slicing
4/5/12
Thinking about Slicing with an Added Perspective
Now that we have both a shell and a disk/washer method for calculating a volume of revolution, we are
in a position where we should ponder which of these two methods will render a more efficient solution –
both are great methods, but it is sensible to pick the method that is simplest to work with.
Before we begin, let us recall what characteristic about our “slice” dictates whether we use disks/washers
or shells to generate a desired solid:
 if a slice to be revolved is perpendicular to the axis of rotation, then the slice will generate a
disk/washer
 if a slice to be revolved is parallel to the axis of rotation, then the slice will generate a shell
Let us use an example to allow us the opportunity to work with both of these methods – we shall then
see the factors we should consider when deciding to use disks/washers vs shells.
Scenario 1a
The area enclosed by the curve y  2 x  x 2 and the x-axis is rotated about the y-axis. Calculate the
volume formed using the disk/washer method -- first set up a Riemann Sum and then an integral
expression, which you can then evaluate to determine the volume.
y

x



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Calculus
Let us now look at the same scenario, but this time, let’s use the shell method to determine the volume of
revolution:
Scenario 1b
The area enclosed by the curve y  2 x  x 2 and the x-axis is rotated about the y-axis. Calculate the
volume formed using the shell method -- first set up a Riemann Sum and then an integral expression,
which you can then evaluate to determine the volume.
y

x




Calculus
Hopefully you are now aware of one reason one might consider setting up a definite integral using
horizontally (dy) vs vertically (dx)-oriented slices. Let’s look at another scenario, different in nature:
Scenario 2a
Suppose we consider the region enclosed by y  x , y  2  x 2 , and the x-axis. If we revolve this area
around the x-axis using vertical “slices”, determine its volume – again, first set up a Riemann Sum and
then an integral expression, which you can then evaluate to determine the volume.
Note: you will first want to determine whether you will be working with disks/washers vs shells.
y

x


Calculus
Scenario 2b
Suppose we consider the region enclosed by y  x , y  2  x 2 , and the x-axis. If we revolve this area
around the x-axis using horizontal “slices”, determine its volume – again, first set up a Riemann Sum
and then an integral expression, which you can then evaluate to determine the volume.
Note: you will first want to determine whether you will be working with disks/washers vs shells.
y

x


Take-Away Question: Based on these two exercises, what factors should one consider when constructing
an expression to determine a volume of revolution?