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