MvCalc20

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We have I=\iint xy(x+y)dxdy\,

Now the order of integration is with respect x and then with respect to y.Hence,we take an elementary strip parallel to the x-axis. The limits for x are easily seen to be from x=y,x={\sqrt  {y}}\,. The limits for y are then from y=0\, to y=1\,. Hence,we write

I=\int _{0}^{1}\int _{{y}}^{{{\sqrt  {y}}}}xy(x+y)dxdy\,

\int _{0}^{1}[{\frac  {x^{3}}{3}}y+{\frac  {x^{2}}{2}}y^{2}]_{{x=y}}^{{{\sqrt  {y}}}}dy\,

=\int _{0}^{1}[{\frac  {y^{{{\frac  {3}{2}}}}}{3}}y+{\frac  {y^{3}}{2}}-{\frac  {y^{4}}{3}}-{\frac  {y^{4}}{2}}]dy\,

=\int _{0}^{1}[{\frac  {y^{{{\frac  {5}{2}}}}}{3}}+{\frac  {y^{3}}{2}}-{\frac  {5}{6}}y^{4}]dy\,

=[{\frac  {2}{7}}{\frac  {y^{{{\frac  {7}{2}}}}}{3}}+{\frac  {y^{4}}{8}}-{\frac  {5}{6}}{\frac  {y^{5}}{5}}]_{0}^{1}\,

={\frac  {2}{21}}+{\frac  {1}{8}}-{\frac  {1}{6}}={\frac  {3}{56}}\,

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