π [4x + 2)^2 − (x^2 + 2)^2 ] dxSuppose instead that the region was revolved about the line x = −2. Sketch the region again.
You’ll have to slice the region horizontally this time; this means you’re going to solve each equation for x
in terms of y: x = and x = . We also need to find the y-coordinates of the intersection of the two
curves: y = 0, 16.
Notice also that, again, each radius is going to be increased by 2 to reflect the shift in the axis of
revolution. Thus, we will have to evaluate the integral.
Finding the volumes isn’t that hard, once you’ve drawn a picture, figured out whether you need to slice
vertically or horizontally, and determined whether the axis of revolution has been shifted. Sometimes,
though, there will be times when you want to slice vertically yet revolve around the y-axis (or slice
horizontally yet revolve about the x-axis). Here’s the method for finding volumes in this way.
CYLINDRICAL SHELLS
Let’s examine the region bounded above by the curve y = 2 − x^2 and below by the curve y = x^2 , from x = 0
to x = 1. Suppose you had to revolve the region about the y-axis instead of the x-axis.