Okay, I did some researching on the almighty Internet and found this...
Draw a 2-dimensional picture of a cross-section through the middle. Look at the right hand side. You have a 2-dimensional object bounded on the right by the curve
and on the left by the line
. The points where these curves intersect are
and
. We want the volume of the solid of revolution formed by rotating this about the y axis.
Method 1 (the shell method):
Slice the 2-dimensional object into vertical strips. Each such vertical strip is located at position x, has width dx, and height
. Rotating this strip about the vertical axis gives a thin cylindrical shell. Slice this open and lay it out flat. The result is a rectangular slap of width
(the circumference of the shell), height
, and thickness dx so
and
Method 2 (the "washer" method):
Slice the 2-dimensional object into horizontal strips. Each such horizontal strip is located at position y and has width dy. Its left endpoint is at
and its right endpoint is at
. Rotating this strip about the vertical axis gives a thin flat "washer" of thickness dy. The inner radius of this (annulus) is
and the outer radius is
. The volume is
I'm sure you know how to solve either of these integrals.