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

\(\mathbf{F}(x, y, z)=\left\langle e^{z^{2}}, 2 y+\sin \left(x^{2} zight), 4 z+\sqrt{x^{2}+9 y^{2}}ightangle, \mathcal{S}\) is the region \(x^{2}+y^{2} \leq z \leq 8-x^{2}-y^{2}\).

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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 =JJJw J F.dS= div(F) dV 1