Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).

\(\mathbf{F}(x, y, z)=\left\langle x, y^{2}, z+yightangle, \mathcal{S}\) is the boundary of the region contained in the cylinder \(x^{2}+y^{2}=4\) between the planes \(z=x\) and \(z=8\).

THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W in R. Assume that S is piecewise smooth and is oriented by normal vectors pointing to the outside of W. If F is a vector field whose components have continuous partial derivatives in an open domain containing W, then J div(F) dV F.dS= 1