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?

Without using polar coordinates, we can evaluate the integral of interest $I$ given by
$$I=\int_{-a}^a \int_{-\sqrt{a^2-y^2}}^{\sqrt{a^2-y^2}}|xy|\,dx\,dy$$
Exploiting even symmetry reveals
$$\begin{align}
I&=4\int_{0}^a \int_0^{\sqrt{a^2-y^2}}xy\,dx\,dy\\\\
&=4\int_0^a y\int_0^{\sqrt{a^2-y^2}}x\,dx\,dy\\\\
&=4\int_0^a y\left(\frac12 (a^2-y^2)\right)\,dy\\\\
&=\int_0^a (a^2-y^2)\,d(y^2) \\\\
&=a^4-\frac12 a^4\\\\
&=\frac12 a^4
\end{align}$$
as expected!
