The problem with a solution appears in M. Moore, __Symmetrical Intersections of right circular cylinders__, *Mathematical Gazette*, No 405, 1974

The answer is x^{3}(sqrt(2) - 1).

I have not seen the paper itself (the reference is from D. Wells' *The Penguin Dictionary of Curious and Interseting Geometry*), but in a recent *Am Math Monthly* (v 111, n 6. Jun-July 2004, pp 496-508) there is an article by T. Apostol and M. Mnatsakanian, *A fresh Look at the Method of Archimedes* that shows an approach to solving a similar problem (intersection of semicylinders.)