VTRMC 2006 Problem 7

Three spheres each of unit radius have centers $P,Q,R$ with the property that the center of each sphere lies on the surface of the other two spheres. Let $C$ denote the cylinder with cross-section $PQR$ (the triangular lamina with vertices $P,Q,R$ ) and axis perpendicular to $PQR$ . Let $M$ denote the space which is common to the three spheres and the cylinder $C$ , and suppose the mass density of $M$ at a given point is the distance of the point from $PQR$ . Determine the mass of $M$ .