VC5.42

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Using the divergence theorm

\iint _{S}(zx^{2}dxdy+x^{3}dydz+yx^{2}dzdx)=\iiint _{V}[{\frac  {\partial }{\partial z}}(zx^{2})+{\frac  {\partial }{\partial x}}(x^{3})+{\frac  {\partial }{\partial y}}(yx^{2})]dV\,

=\iiint _{V}(x^{2}+3x^{2}+x^{2})dV\,

=\int _{{z=0}}^{{3}}\int _{{y=-2}}^{{2}}\int _{{x=-{\sqrt  {2^{2}-y^{2}}}}}^{{{\sqrt  {2^{2}-y^{2}}}}}x^{2}dxdydz\,

=5\int _{{z=0}}^{{3}}\int _{{y=-2}}^{{2}}[{\frac  {x^{3}}{3}}]_{{x=-{\sqrt  {2^{2}-y^{2}}}}}^{{{\sqrt  {2^{2}-y^{2}}}}}dydz\,

={\frac  {10}{3}}\int _{0}^{3}\int _{{-2}}^{{2}}(2^{2}-y^{2})^{{{\frac  {3}{2}}}}dydz\,

={\frac  {10}{3}}\int _{{-2}}^{{2}}[z(2^{2}-y^{2})^{{{\frac  {3}{2}}}}dy\,

=10\times 2\int _{0}^{2}(2^{2}-y^{2})^{{{\frac  {3}{2}}}}dy\,

=20\int _{{0}}^{{{\frac  {\pi }{2}}}}8(\cos ^{3}\theta )(2\cos \theta )d\theta \, [on puttingy=2\sin \theta ,dy=2\cos \theta d\theta \,]

=320\int _{{0}}^{{{\frac  {\pi }{2}}}}\cos ^{4}\theta d\theta =320\times {\frac  {3}{4}}{\frac  {1}{2}}{\frac  {\pi }{2}}=60\pi \,

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