Calculating area, using integrals.

Question

How to calculate an area, bounded by:

$$ x^{4} + y^{4} = 2a^2xy $$

I am quite good in solving multiple integrals, but i don't know how to proceed to an integral from here.

Answer 1

I would use polar coordinates $x=r\cos\theta$ and $y=r\sin\theta$. The integrand $dxdy$ will become $rdrd\theta$. The boundary equation is $$r^4(\cos^4\theta + \sin^4\theta)=2a^2r^2\sin\theta\cos\theta$$ With some simple manipulation, you get $$r^2=\frac{a^2\sin(2\theta)}{1-\sin^2(2\theta)/2}$$