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

\(\mathbf{F}(x, y, z)=\left\langle z x, y x^{3}, x^{2} zightangle, \mathcal{S}\) is the surface that bounds the solid region with boundary given by \(y=\) \(4-x^{2}-z^{2}, y=0\).

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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 1 F.dS= div(F) dV 1