VC5.40

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Here,{\mathrm  {div}}F={\frac  {\partial }{\partial x}}(xy^{2})+{\frac  {\partial }{\partial y}}(y^{3})+{\frac  {\partial }{\partial z}}(y^{2}z)=y^{2}+3y^{2}+y^{2}=5y^{2}\, --(1)

BY divergence theorm,\iint _{S}F\cdot ndS=\iiint _{V}{\mathrm  {div}}FdV\,,V being volume enclosed by S.

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

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

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

=10\int _{{-3}}^{{3}}y^{2}{\sqrt  {3^{2}-y^{2}}}[z]_{0}^{2}dy=10\int _{{-3}}^{{3}}2y^{2}{\sqrt  {3^{2}-y^{2}}}dy\,

=40\int _{0}^{3}y^{2}{\sqrt  {3^{2}-y^{2}}}dy\, [Since integrand is even function]

=40\int _{{0}}^{{{\frac  {\pi }{2}}}}(9\sin ^{2}\theta )(3\cos \theta )(3\cos \theta )d\theta \,,on putting y=3\sin \theta ,dy=3\cos \theta d\theta \,

=810\int _{{0}}^{{{\frac  {\pi }{2}}}}\sin ^{2}2\theta d\theta =810\int _{{0}}^{{{\frac  {\pi }{2}}}}[{\frac  {1-\cos 4\theta }{2}}d\theta =405[\theta -{\frac  {\sin 4\theta }{2}}]_{{0}}^{{{\frac  {\pi }{2}}}}\,

={\frac  {405\pi }{2}}\,

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