Integrate f(x, y,z)=12xz over the region in the first octant above the parabolic cylinder z=y^2 and below the paraboloid z=8−2x^2−y^2.

The intersection of the two surfaces occurs along the cylinder , so we have the - and -coordinates of the points in the bounded region contained within and , while .

The triple integral is then given by



where denotes the bounded region in the first quadrant.

Integrating in Cartesian coordinates is easy enough to do in this order.