MvCalc40

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The area of the plane A=\iint _{D}dxdy\, where D is the domain formed by 3y^{2}=25x,5x^{2}=9y\,

The limits from the given equations are: x from 0 to 3 and y from {\frac  {5}{9}}x^{2}\, to {\frac  {5{\sqrt  {x}}}{{\sqrt  {3}}}}\,

Now the area A=\int _{0}^{3}[\int _{{{\frac  {5}{9}}x^{2}}}^{{{\frac  {5{\sqrt  {x}}}{{\sqrt  {3}}}}}}dy]dx\,

=\int _{0}^{3}[x]_{{{\frac  {5}{9}}x^{2}}}^{{{\frac  {5{\sqrt  {x}}}{{\sqrt  {3}}}}}}dx\,

=\int _{0}^{3}[{\frac  {5{\sqrt  {x}}}{{\sqrt  {3}}}}-{\frac  {5}{9}}x^{2}]dx\,

=[{\frac  {5}{{\sqrt  {3}}}}{\frac  {2}{3}}x^{{{\frac  {3}{2}}}}-{\frac  {5}{27}}x^{3}]_{0}^{3}\,

={\frac  {10}{3{\sqrt  {3}}}}(3^{{{\frac  {3}{2}}}})-{\frac  {5}{27}}(27)\,

=10-5=5\, on simplification.

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