MvCalc19

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The order of integration is first with respect to x. Hence we consider the strip parallel to x-axis.

A strip extends from x=0 to {\sqrt  {a^{2}-y^{2}}}\,

The limits of y are from 0 to a.

Therefore,the integral is I=\int _{0}^{a}\int _{{0}}^{{{\sqrt  {a^{2}-y^{2}}}}}xydxdy\,

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

={\frac  {1}{2}}\int _{0}^{a}(a^{2}-x^{2})ydy\,

={\frac  {1}{2}}[a^{2}{\frac  {y^{2}}{2}}-{\frac  {y^{4}}{4}}]_{0}^{a}\,

={\frac  {1}{2}}[{\frac  {a^{4}}{2}}-{\frac  {a^{4}}{4}}]={\frac  {a^{4}}{8}}\,

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