VC5.37

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Using the Gauss divergence theorm,we have

\iint _{S}[x^{2}dydz+y^{2}dzdx+2z(xy-x-y)dxdy]=\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}xydxdydz\,

=2\int _{{z=0}}^{{1}}\int _{{y=0}}^{{1}}\int {x=0}^{{1}}xydxdydz\,

=2\int _{{z=0}}^{{1}}\int _{{y=0}}^{{1}}[{\frac  {x^{2}y}{2}}]_{0}^{1}dydz\,

=2\int _{{z=0}}^{{1}}\int _{{y=0}}^{{1}}{\frac  {1}{2}}dydz\,

=\int _{{z=0}}^{{1}}\int _{{y=0}}^{{1}}ydydz\,

=\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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