Integrate f(x, y,z)=x over the region in the first octant (x> 0, y> 0, z> 0) above z=y^2 and below z=8-2x^2-y^2

which means the intersection of the parabolic cylinder  and paraboloid is a circle of radius 2 centered at the origin.

So the integral can be represented in Cartesian coordinates by



where is the region between the two surfaces.

Converting to cylindrical coordinates will make this slightly easier to compute.




Letting denote the same region in cylindrical coordinates, you have



In either case the integral reduces to 0.

52 or what they said ☝️