The problem with a solution appears in

M. Moore, Symmetrical Intersections of right circular cylinders, Mathematical Gazette, No 405, 1974

The answer is x3(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.)