Double Integral Help??!?

It always helps to draw a picture. What I see is a cylinder (x2+y2=4 in 3 dimensions) ping through a sphere. I would think the outside is defined by the equation of the sphere and the inside by the equation of the cylinder. I think you have to slice the cylinder up because of the integration of a circle, being a double valued gizmo, gives you a value of zero. Thats where the 2 comes in. Good luck.