The wrong way of finding the average distance between two points on a circle I was trying to find the average distance between two points on a circle and got the following result. 
Why is my method wrong?

Your $P_1 P_2$ value is correct but you are integrating over y whereas you have to integrate it over the arc between y = 0 and 2r. So the best way is to do it in polar coordinates.
Coordinates of a point on the circle is $(r\sin2\alpha, r - r\cos2\alpha)$ where $\alpha$ is the angle between x-axis and line $P_1P_2$.
Sum of all lines from the origin to the points on circle till $y = 2r$ is
= $r\sqrt2 \int_0^{\pi/2} \sqrt{1-\cos2\alpha} \space d\alpha$, using $P_1P_2 = \sqrt{2ry}$ as per your work.
= $r\sqrt2 \int_0^{\pi/2} \sqrt2 \sin\alpha\space d\alpha = 2r[-cos\alpha]_0^{\pi/2} = 2r$
This is over angle $\pi/2$. So the average length of $P_1P_2 = \dfrac{4r}{\pi}$
