The radius of each cylinder is increased by 1 because of the shift in the axis of revolution, so the integral
looks like the following:
2 π (y + 1)(y − y^3 ) dyWasn’t this fun? Volumes of Solids of Revolution require you to sketch the region carefully and to decide
whether it’ll be easier to slice the region vertically or horizontally. Once you figure out the slices’
boundaries and the limits of integration (and you’ve adjusted for an axis of revolution, if necessary), it’s
just a matter of plugging into the integral. Usually, you won’t be asked to evaluate the integral unless it’s a
simple one. Once you’ve conquered this topic, you’re ready for anything.
VOLUMES OF SOLIDS WITH KNOWN CROSS-SECTIONS
There is one other type of volume that you need to be able to find. Sometimes, you will be given an object
where you know the shape of the base and where perpendicular cross-sections are all the same regular,
planar geometric shape. These sound hard, but are actually quite straightforward. This is easiest to
explain through an example.
Example 8: Suppose we are asked to find the volume of a solid whose base is the circle x^2 + y^2 = 4, and
where cross-sections perpendicular to the x-axis are all squares whose sides lie on the base of the circle.
How would we find the volume?
First, make a drawing of the circle.