Over a curve $C$ given by $(x^2+y^2)^2=30^2(x^2-y^2)$, What is $ \oint\limits_C |y|\,\mathrm ds. $
I've tried working on it but I couldn't get the solution.
Here's how I did it:
Using polar coordinates $ \begin{cases} x(t) &= 30 \sqrt{\cos 2t}\cdot\cos t \\ y(t) &= 30 \sqrt{\cos 2t} \cdot\sin t \end{cases} $ thereafter, $\mathrm ds = 30 \sqrt{ \sec 2t}\cdot \mathrm dt$.
Finally, integral over $ \oint\limits_C |y|\,ds = 2 \cdot 30^2\int\limits_{-\pi/4}^{\pi/4}\sqrt{\cos 2t} \cdot\sqrt{\sec 2t} \sin t\, \mathrm dt = 0. $
I've spent many hours on this problem already. Will someone be kind enough to please help me? Thanks.