MvCalc47

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Here f(x,y)={\frac  {x^{2}+y^{2}}{a}}\,

V=\iint _{R}f(x,y)dxdy\,

=\int _{0}^{a}\int _{{-{\sqrt  {a^{2}-x^{2}}}}}^{{{\sqrt  {a^{2}-x^{2}}}}}{\frac  {x^{2}+y^{2}}{a}}dydx\,

Changing these into polar,that is x=r\cos \theta ,y=r\sin \theta ,x^{2}+y^{2}=r^{2}\,

The base R is covered by varying r from 0 to a and theta from 0 to 2\pi \,

V=\int _{{0}}^{{2\pi }}\int _{0}^{a}({\frac  {r^{2}}{a}}rdrd\theta \,

=\int _{{0}}^{{2\pi }}[{\frac  {r^{4}}{4a}}]_{0}^{a}d\theta ={\frac  {a^{3}}{4}}2\pi ={\frac  {\pi a^{3}}{2}}\,

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