VC5.62

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Using the divergence theorm,we get

\iint _{S}(x^{3}dydz+y^{3}dzdx+0dxdy)=\iiint _{V}[{\frac  {\partial }{\partial x}}(x^{3})+{\frac  {\partial }{\partial y}}(y^{3})+{\frac  {\partial }{\partial z}}(0)]dxdydz\,

=3\iiint _{V}(x^{2}+y^{2})dxdydz={\frac  {3}{\rho }}\iiint _{v}(x^{2}+y^{2})dm\,

[Here dm=\rho dxdydz\,,rho being constant density of material of sphere]

={\frac  {3}{\rho }}\times (momentofinertiaofthegivensphereaboutz-axis)\,

={\frac  {3}{\rho }}\times [{\frac  {2}{5}}Ma^{2}]\, where M is the mass of the sphere.

={\frac  {3}{\rho }}\times {\frac  {2}{5}}\times ({\frac  {4}{3}}\pi a^{3}\rho )a^{2}\, [As M={\frac  {4}{3}}\pi a^{3}\rho \,]

={\frac  {8}{5}}\pi a^{5}\,

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