VC5.37

From Exampleproblems

Using the Gauss divergence theorm,we have

$\iint_S[x^2 dy dz+y^2 dz dx+2z(xy-x-y)dx dy]=\iiint_V[\frac{\partial}{\partial x}(x^2)+\frac{\partial}{\partial y}(y^2)+\frac{\partial}{\partial z}(2z(xy-x-y))]dV\,$

= $\iiint_V(2x+2y+2xy-2x-2y)dV=2\iiint_V xy dx dy dz\,$

= $2\int_{z=0}^{1}\int_{y=0}^{1}\int{x=0}^{1} xydx dy dz\,$

= $2\int_{z=0}^{1}\int_{y=0}^{1}[\frac{x^2y}{2}]_0^1 dy dz\,$

= $2\int_{z=0}^{1}\int_{y=0}^{1}\frac{1}{2}dy dz\,$

= $\int_{z=0}^{1}\int_{y=0}^{1}ydy dz\,$

= $\int{z=0}^{1}[\frac{y^2}{2}]_0^1 dz=\frac{1}{2}\int_0^1 dz=\frac{1}{2}\,$

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VC5.37

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