MvCalc61

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In cylindrical coordinates,the equation of the sphere becomes r^{2}+z^{2}=a^{2}\,

and the volume element is rdrd\theta dz\,,then the volume is given by

V=\int _{{0}}^{{2\pi }}\int _{0}^{a}\int _{{-{\sqrt  {a^{2}-r^{2}}}}}^{{{\sqrt  {a^{2}-r^{2}}}}}rdrd\theta dz\,

=2\int _{{0}}^{{2\pi }}\int _{0}^{a}{\sqrt  {a^{2}-r^{2}}}rdrd\theta \,

=\int _{{0}}^{{2\pi }}\int _{{0}}^{{a^{2}}}{\sqrt  {a^{2}-R}}dRd\theta \,

=\int _{{0}}^{{2\pi }}[-{\frac  {2(a^{2}-R)^{{{\frac  {3}{2}}}}}{3}}]_{{0}}^{{a^{2}}}d\theta \,

=-{\frac  {2}{3}}(2\pi )(-a^{3})={\frac  {4}{3}}\pi a^{3}\,

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