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
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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