That's an interesting question. In order to avoid confusing myself, I'm going to start with a much simplified version of that problem and then work my way upwards: Suppose they only wanted you to calculate the volume of the solid formed when the area between the curve y = 2 - x^2 and the lines x = 0 and x = 1 was rotated around the y-axis.
How would I calculate that volume?
Break it down into two parts:
The bottom half 0<x<1, 0<y<1, is just a simple cylinder. Volume is thus
The top half can be constructed from a series of thin horizontal disks. Volume of each of these disks are
, thus we must integrate over y.
We know that:
- the region we're integrating is from y=1 to y=2 (this is the curved part)
-
Thus,
Total volume is thus