Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate the flux of F across S. Fx, y, z = 2x3 + y3i + y3 + z3j + 3y2zk, S is the surface of the solid bounded by the paraboloid z = 1 − x2 − y2 and the xy-plane.

\(\vec F(x,y,z)=(2x^3+y^3)\,\vec\imath+(y^3+z^3)\,\vec\jmath+3y^2z\,\vec k\)has divergence\(\mathrm{}div{}\vec F(x,y,z)=6x^2+3y^2+3y^2=6(x^2+y^2)\)Then by the divergence theorem, the flux of \(\vec F\) across \(S\) is equal to the integral of \(\mathrm{}div{}\vec F\) over the interior of \(S\). In cylindrical coordinates, this integral is\(\displaystyle6\int_0^{}2\pi{}\int_0^1\int_0^{}1-r^2{}r^3\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\boxed{}\pi{}\)