find the area bounded by y^(2/3)+x^(2/3)=a^(2/3) and co-ordinate axes.

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- Nov 17th 2010, 07:20 AMayushdadhwalarea
find the area bounded by y^(2/3)+x^(2/3)=a^(2/3) and co-ordinate axes.

- Nov 17th 2010, 07:33 AMHallsofIvy
- Nov 17th 2010, 08:40 AMTheEmptySet
Use the parametrization $\displaystyle x=a\cos^3(t) $ and $\displaystyle y=a\sin^3(t)$ to complete the simple close.

By Greene's theorem we get

$\displaystyle \frac{1}{4}\iint_DdA=-\frac{1}{4}\oint y\,dx=-a^2\int_{0}^{2\pi}\sin^3(t)(3\cos(t)\sin(t))dt=-\frac{3a^2}{4}\int_{0}^{2\pi}\sin^{4}(t)\cos(t)\,d t $