Planes and Surfaces

The coordinate change $2x-y=a,x+y-1=b$ will affect the volume by a factor of $\det\begin{bmatrix}2&-1\\1&1\end{bmatrix}=3$ . The volume of the solid paraboloid enclosed by the plane $z=4$ and the surface $z=a^2+b^2$ is half of the volume of the circumscribed cylinder, which is $\frac{1}{2}\times 4\times 4\pi=8\pi$ . Thus the volume of the given solid region is $\frac{8\pi}{3}$ and the answer is $3+8=\boxed{11}$ .