VC5.35

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Let V be the volume enclosed by S. Then,using Gauss divergence theorm,we have

\iint _{S}[(x^{3}-yz)dzdx+(-2x^{2}y)dzdx+zdxdy]\,

=\iiint _{V}[{\frac  {\partial }{\partial }}(x^{3}-yz)+{\frac  {\partial }{\partial y}}(-2x^{2}y)+{\frac  {\partial }{\partial z}}(z)]dxdydz\,

=\iiint _{V}(3x^{2}-2x^{2}+1)dxdydz=\int _{{z=0}}^{{a}}\int _{{y=0}}^{{a}}\int _{{x=0}}^{{a}}(x^{2}+1)dxdydz\,

=\int _{{z=0}}^{{a}}\int _{{y=0}}^{{a}}[{\frac  {x^{3}}{3}}+x]_{0}^{a}dydz\,,first integrating with x.

=\int _{{z=0}}^{{a}}\int _{{y=0}}^{{a}}[{\frac  {a^{3}}{3}}+a]dydz\,

=[{\frac  {a^{3}}{3}}+a]\int _{{z=0}}^{{a}}\int _{{y=0}}^{{a}}dydz=a[{\frac  {a^{3}}{3}}+a]\int _{0}^{a}dz\,

=a^{2}[{\frac  {a^{3}}{3}}+a]\,

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