VC5.60

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\iint _{S}(x^{2}i+y^{2}j+z^{2}k)\cdot ndS=\iiint _{V}(x^{2}i+y^{2}j+z^{2}k)dV\,,V being volume enclosed by S.

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

=2\int _{{z=-c}}^{{c}}\int _{{y=-b{\sqrt  {(1-{\frac  {z^{2}}{c^{2}}})}}}}^{{b{\sqrt  {(1-{\frac  {z^{2}}{c^{2}}})}}}}\int _{{x=-a{\sqrt  {(1-{\frac  {y^{2}}{b^{2}}}-{\frac  {z^{2}}{c^{2}}})}}}}^{{a{\sqrt  {(1-{\frac  {y^{2}}{b^{2}}}-{\frac  {z^{2}}{c^{2}}})}}}}(x+y+z)dxdydz\,

=2\int _{{z=-c}}^{{c}}\int _{{y=-b{\sqrt  {(1-{\frac  {z^{2}}{c^{2}}})}}}}^{{b{\sqrt  {(1-{\frac  {z^{2}}{c^{2}}})}}}}[{\frac  {x^{2}}{2}}+x(y+z)]_{{x=-a{\sqrt  {(1-{\frac  {y^{2}}{b^{2}}}-{\frac  {z^{2}}{c^{2}}})}}}}^{{a{\sqrt  {(1-{\frac  {y^{2}}{b^{2}}}-{\frac  {z^{2}}{c^{2}}})}}}}dydz\,

=4\int _{{z=-c}}^{{c}}\int _{{y=-b{\sqrt  {(1-{\frac  {z^{2}}{c^{2}}})}}}}^{{b{\sqrt  {(1-{\frac  {z^{2}}{c^{2}}})}}}}(y+z){\sqrt  {[1-{\frac  {y^{2}}{b^{2}}}-{\frac  {z^{2}}{c^{2}}}]}}dydz\,

=8a\int _{{z=-c}}^{{c}}\int _{{y=0}}^{{b{\sqrt  {(1-{\frac  {z^{2}}{c^{2}}})}}}}z[1-{\frac  {y^{2}}{b^{2}}}-{\frac  {z^{2}}{c^{2}}}]^{{{\frac  {1}{2}}}}dydz\,

={\frac  {8a}{b}}\int _{{z=-c}}^{{c}}z[{\frac  {y}{2}}(b^{2}(b-{\frac  {z^{2}}{c^{2}}})-y^{2})^{{{\frac  {1}{2}}}}+{\frac  {b^{2}}{2}}(1-{\frac  {z^{2}}{c^{2}}})\arcsin({\frac  {y}{b{\sqrt  {1-{\frac  {z^{2}}{c^{2}}}}}}}]_{{y=0}}^{{b{\sqrt  {1-{\frac  {z^{2}}{c^{2}}}}}}}dz\,

={\frac  {8a}{b}}\int _{{-c}}^{{c}}z[{\frac  {b^{2}}{2}}(1-{\frac  {z^{2}}{c^{2}}})\arcsin(1)]dz\,

={\frac  {8a}{b}}\times {\frac  {b^{2}\pi }{4}}\int _{{-c}}^{{c}}z[1-{\frac  {z^{2}}{c^{2}}}]dz=0\, [Since z(1-{\frac  {z^{2}}{c^{2}}})\, is an odd function]

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