I am solving the problem as follows:
"Find the average distance from a point $P(x,y)$ in the disk $x^2+y^2\le a^2$ to the origin."
The radius of the disk is $a$ and the area of the disk is $a^2\pi$.
The function describing the distance between the point in the disc and the origin would be
$$f(x,y)=\sqrt{(x-0)^2+(y-0)^2}$$
Then we can convert the distance formula into polar form
$$g(r,\theta)=\sqrt{r^2}=r$$
Then the average distance is
$$\frac{1}{a^2\pi}\iint\limits_{R}g(r,\theta)rdrd\theta=\frac{1}{a^2\pi}\iint\limits_{R}r^2drd\theta$$
Now consider the limits.
$$0\le x^2+y^2\le a^2\implies 0\le r\le a$$
But I am not sure about the limits of $\theta$.
Should it be
$$0\le\theta\le\pi$$
Or $$0\le\theta\le 2\pi?$$