The integral is:
$$\iint\limits_{\Omega}|xy|\,\mathrm{d}x\,\mathrm{d}y$$
$\Omega$ is given by the circle with radius $a$ and center in the zero of coordinat plane. I am confused what to do with mudule. And how many integration regions should I have? As it is module I guess I have to take $1/4$ of circle.
This integral is four times that over the portion of the disc $A$ in the first quadrant. This is $$4\int_A xy\,dx\,dy.$$ This is easily done via polar coordinates $x=r\cos t$, $y=r\sin t$ giving $$4\int_0^a\int_0^{\pi/2}(r\cos t)(r\sin t)r\,dt\,dr.$$