Green's formula for area

Question

Using Green theorem I need to calculate the area bounded with $(x+y)^4=x^2y$. First(after converting to polar coordinates) i get $x=\cos^6\phi\sin^2\phi$ and $y=\cos^4\phi\sin^4\phi$. And after i plug that into Green's formula i get $\int\limits_{0}^{2\pi} \cos^9\phi\sin^5\phi d\phi$ and that is $0$. But when i set bounds for $\phi$ to be $0, \frac{\pi}{2}$ i get correct solution. Why we have $\frac{\pi}{2}$ for $\phi$ when we don't have any restrictions?

Answer 1

The closed curve is drawn once as $\phi$ goes from $0$ to $\pi/2$, because both the functions for $x$ and $y$ have period $\pi/2$. Then as $\phi$ goes from $\pi/2$ to $\pi$ the same curve is traced backward. So integrating from $0$ to $\pi$ should give $0$. This act is repeated from $\pi$ to $2\pi$.