VC5.36

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

\iint _{S}[(x^{3}-yz)i-2x^{2}yj+2k]\cdot ndS=\iiint _{V}{\mathrm  {div}}[(x^{3}-yz)i-2x^{2}yj+2k]dV\,

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

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

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

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

={\frac  {a^{4}}{3}}\int _{0}^{a}dz={\frac  {a^{5}}{3}}\,

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