MvCalc11

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I=\int _{0}^{a}[\int _{{0}}^{{{\sqrt  {a^{2}-x^{2}}}}}y^{3}dy]dx\,

\int _{{0}}^{{{\sqrt  {a^{2}-x^{2}}}}}y^{3}dy=[{\frac  {y^{4}}{4}}]_{{0}}^{{{\sqrt  {a^{2}-x^{2}}}}}\,

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

={\frac  {a^{4}-2a^{2}x^{2}+x^{4}}{4}}\,

Now I=\int _{0}^{a}[{\frac  {a^{4}-2a^{2}x^{2}+x^{4}}{4}}]dx\,

=\int _{0}^{a}{\frac  {a^{4}}{4}}dx-\int _{0}^{a}{\frac  {a^{2}x^{2}}{2}}dx+\int _{0}^{a}{\frac  {x^{4}}{4}}dx\,

={\frac  {a^{4}}{4}}[x]_{0}^{a}-{\frac  {a^{2}}{2}}[{\frac  {x^{3}}{3}}]_{0}^{a}+[{\frac  {x^{5}}{20}}]_{0}^{a}\,

={\frac  {a^{5}}{4}}-{\frac  {a^{5}}{6}}+{\frac  {a^{5}}{20}}\,

={\frac  {15a^{5}-10a^{5}+3a^{5}}{60}}={\frac  {8a^{5}}{60}}={\frac  {2a^{5}}{15}}\,

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