VC5.41

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Let V be the volume enclosed by closed surface S of given cylinder bounded by (say)planes OAB(x=0) and O'A'B'(x=2)lying in the first octant.

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

=\iiint _{V}[{\frac  {\partial }{\partial x}}(2x^{2}y)+{\frac  {\partial }{\partial y}}(-y^{2})+{\frac  {\partial }{\partial z}}(4xz^{2})]dV=\iiint _{V}(4xy-2y+8xz)dV\,

=2\int _{{x=0}}^{{2}}\int _{{z=0}}^{{3}}\int _{{y=0}}^{{{\sqrt  {9-z^{2}}}}}(2xy-y+4xz)dxdydz\,

=2\int _{{x=0}}^{{2}}\int _{{z=0}}^{{3}}[xy^{2}-{\frac  {y^{2}}{2}}+4xyz]_{{y=0}}^{{{\sqrt  {9-z^{2}}}}}dxdz\,

=2\int _{{x=0}}^{{2}}\int _{{z=0}}^{{3}}[x(9-z^{2})-{\frac  {x}{2}}(9-z^{2})+4xz{\sqrt  {9-z^{2}}}]_{{x=0}}^{{2}}dz\,

=2\int _{0}^{3}[2(9-z^{2})-(9-z^{2})+8z{\sqrt  {9-z^{2}}}]dz\,

=2\int _{0}^{3}[9-z^{2}-4(9-z^{2})^{{{\frac  {1}{2}}}}(-2z)]dz\,

=2[9z-{\frac  {z^{2}}{3}}-4\cdot {\frac  {2}{3}}(9-z^{2})^{{{\frac  {3}{2}}}}]_{0}^{3}=2[27-9+{\frac  {8}{3}}\cdot 27]=180\,

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