MvCalc39

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Area of the plane A=\iint _{D}dxdy\,

The limits for the integration are y-y^{2}+y=0,y^{2}=2y,y=2\, the limits of y are from 0 to 2.The limits of x are from -y\, to y-y^{2}\,

Now the area A=\int _{0}^{2}[\int _{{-y}}^{{y-y^{2}}}dx]dy\,

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

=\int _{0}^{2}[y-y^{2}+y]dy=\int _{0}^{2}(2y-y^{2})dy\,

=[y^{2}-{\frac  {y^{3}}{3}}]_{0}^{2}=4-{\frac  {8}{3}}={\frac  {4}{3}}\,

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