Verify Gauss divergence theorem, for $$\iint_S xz dydz + x^2y dzdx + y^2z dxdy$$ where $S$ is the outer side of the surface in the first octant formed by the cylinder $x^2+y^2=1$, the paraboloid of revolution $z=x^2+y^2$, and the coordinate planes.
Considering $\vec F=xz\,i+x^2y\,j+y^2z\,k$, we have $div \vec F=z+x^2+y^2.$
By Gauss divergence theorem, $$\iint_S\vec F.d\vec S=\iiint div \vec F\, dV=\iiint (z+x^2+y^2) dxdydz $$
Suggest the limits and remaining part, please.
