VC5.34

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Let V be the volume enclosed by S,then by Gauss divergence theorm,we have

\iint _{S}[4xzdydz+(-y^{2})dzdx+yzdxdy]=\iiint _{V}[{\frac  {\partial }{\partial x}}(4xz)+{\frac  {\partial }{\partial y}}(-y^{2})+{\frac  {\partial }{\partial z}}(yz)]dV\,

=\iiint _{V}(4z-y)dxdydz\,

=\int _{{x=0}}^{{1}}\int _{{y=0}}^{{1}}\int _{{z=0}}^{{1}}(4z-y)dxdydz\,

=\int _{{x=0}}^{{1}}\int _{{y=0}}^{{1}}[2z^{2}-yz]_{{z=0}}^{{1}}dxdy\,

=\int _{{x=0}}^{{1}}\int _{{y=0}}^{{1}}(2-y)dxdy\,

=\int _{{x=0}}^{{1}}[2y-{\frac  {y^{2}}{2}}]_{{y=0}}^{{1}}dx=\int _{{x=0}}^{{1}}[2-{\frac  {1}{2}}]dx={\frac  {3}{2}}\int _{0}^{1}dx={\frac  {3}{2}}\,

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