Integrating modular function $f(x,y)=|xy|$ in the area of the circle $x^2+y^2=a^2$ I need to solve the integral $$\iint_\omega |xy|\,dx\,dy$$ where $\omega:x^2+y^2=a^2$ I created the following integral and I have no idea how can I integrate modular function:
$$\int_{-a}^a dy\int_{-\sqrt{a^2-y^2}}^\sqrt{a^2-y^2} |xy|\,dx$$The only thing that I do know is that $\int |x|\,dx=\frac{x|x|}{2}+C$, but later on I face the next problem: $$\int_{-a}^a|y|\cdot|\sqrt{a^2-y^2}|\cdot \sqrt{a^2-y^2}\,dy$$
Could anyone help me?

You can break your integral into four parts (corresponding to four quadrants) and can integrate easily using polar coordinates.
$$\iint_\omega |xy|dxdy = \int_{r=0}^a\int_{\theta = 0}^{2\pi} f(r, \theta) \ r\  dr\  d\theta$$
$$= \int_{r=0}^a\int_{\theta = 0}^{\pi/2 } r^3 cos\theta \ sin \theta \ dr \ d\theta \ + $$
$$ \quad \ \int_{r=0}^a\int_{\theta = \pi/2}^{\pi } r^3(- cos\theta \ sin \theta) \ dr \ d\theta \ + $$
$$ \int_{r=0}^a\int_{\theta = \pi}^{3\pi/2 } r^3 cos\theta \ sin \theta \ dr \ d\theta \ + $$
$$ \int_{r=0}^a\int_{\theta = 3\pi/2}^{2\pi } r^3(- cos\theta \ sin \theta) \ dr \ d\theta \  $$
