The computation of the volumes of spheres can be done without actually doing any integrals. We only need to know the recusion that B(n) = S(n-1)/n and S(n)=2π B(n-1) and the induction assumption S(0)=2 and B(0)=1. The first recursion follows from integrating the sphere volume up (make shells). The second is an Archimedes trick: the cylinder surface area enclosing a sphere is the same than the surface area because there is an area preserving map from the sphere to the cylinder (project from the center ball which is in the case n=2 the 1-ball (the z axes in the case of Archimedes).