MvCalc62

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Using the spherical polar coordinates,the given integral transforms into

\int _{{0}}^{{{\frac  {\pi }{2}}}}\int _{{0}}^{{{\frac  {\pi }{2}}}}\int _{0}^{a}(r\cos \theta \sin \phi )(r\sin \theta \sin \phi )(r\cos \phi )r^{2}\sin \phi drd\theta d\phi \,

=\int _{{0}}^{{{\frac  {\pi }{2}}}}\int _{{0}}^{{{\frac  {\pi }{2}}}}\int _{0}^{a}r^{5}\cos \theta \sin \theta \sin ^{3}\phi \cos \phi d\phi d\theta dr\,

=[{\frac  {r^{6}}{6}}]_{0}^{a}[{\frac  {\sin ^{2}\theta }{2}}]_{{0}}^{{{\frac  {\pi }{2}}}}[{\frac  {\sin ^{4}\phi }{4}}]_{{0}}^{{{\frac  {\pi }{2}}}}\,

={\frac  {1}{6}}a^{6}{\frac  {1}{2}}{\frac  {1}{4}}\,

={\frac  {a^{6}}{48}}\,

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