Section 5.2, Question 10

If we slice this body using horizontal planes, each section comprises two concentric circles and we want to calculate the area between these circles. The outer circle has radius 1. If the slice is at height y above the origin, then the radius of the inner circle is x where y=2x-x²$. Solving this gives

Of course, we know that x<1, so

So the area of the section is

and therefore, the required volume is given by

The latter part of this integral can be evaluated using the substitution u=1-y. We find that the volume is