Putnam 2006 
Topic: Calculus Geometry

Problem: Find the volume of the region of points $(x,y,z)$ such that \[ (x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2). \]

Solution: We change to cylindrical coordinates, i.e., we put $r = \sqrt{x^2 + y^2}$. Then the given inequality is equivalent to \[ r^2 + z^2 + 8 \leq 6r, \] or \[ (r-3)^2 + z^2 \leq 1. \] This defines a solid of revolution (a solid torus); the area being rotated is the disc $(x-3)^2 + z^2 \leq 1$ in the $xz$-plane. By Pappus's theorem, the volume of this equals the area of this disc, which is $\pi$, times the distance through which the center of mass is being rotated, which is $(2\pi)3$. That is, the total volume is $\boxed{6 \pi^2}$.