Volume as a Surface Integral Let (mathbf{F}(x, y, z)=langle x, y, zangle). Prove that if (mathcal{W}) is

Question:

Volume as a Surface Integral Let $\mathbf{F}(x, y, z)=\langle x, y, zangle$. Prove that if $\mathcal{W}$ is a region in $\mathbf{R}^{3}$ with a smooth boundary $\mathcal{S}$, then
\[
\operatorname{volume}(\mathcal{W})=\frac{1}{3} \iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}
\]

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