Section 5.2, Question 62

Horizontal slices of the horizontal cylinder are rectangles, whereas horizontal slices of the vertical cylinder are circles. So the slices of the part common to both cylinders are intersections of rectangles and circles. They look like

If the slice is at height x above the origin then the length of the straight edge is 2x and the radius of the circle is r. So the area of this shape is

Consequently, the required volume is

where we have used the substitutions u=x/r and v=r2 -x2. The second of these integrals looks rather problematic given the very few techniques of integration currently at our disposal. On the other hand, we see that this integral represents the area under the graph of cosine between x=0 and x=Pi/2, if we integrate along the y-axis. Of course, we get the same answer if we integrate along the x-axis. But now the integrand is cos x which is easy to handle. So we have