1. The problem statement, all variables and given/known data OK, I thought once I knew what the question was asking I'd be able to do it. I was wrong! Consider the volume V inside the cylinder x2 +y2 = 4R2 and between z = (x2 + 3y2)/R and the (x,y) plane, where x, y, z are Cartesian coordinates and R is a constant. Write down a triple integral for the volume V using cylindrical coordinates. Include the limits of integration (three upper and three lower). Evaluate the integral to determine the volume V in terms of R. 2. Relevant equations 3. The attempt at a solution I've sketched both equations, so I have a vague idea of what's going on. The z=... equation is an 'elliptic paraboloid', the other eqn a cylinder. I could find the volume by finding the volume of the cylinder and subtracting the volume above the paraboloid, although the limits for that might be complicated. That's what I'd do in 2D, never done a 3D question. I think I have to find the z coordinate where they intersect, but I don't know how. And I'm supposed to convert to cylindrical, which is probably the only bit I can at least try, using x=rcos(θ), y=rsin(θ) and z=z: r2cos2(θ) + r2sin2(θ) = 4R2 And (r2cos2(θ) + 3r2sin2(θ) )/R = z I think I could replace r with 2R? Then the equations become cos2(θ) + sin2(θ) = 1 And 4Rcos2(θ) + 12Rsin2(θ)=z Not sure where to go now!