why this integral represents the average distance between two point on a unit square

Question

I'm watching this video and get confusing about the integral. Can anyone explain why this integral represents the average distance between two point on a unit square?

Answer 1

You integrate over all possible values of point $1$ ($x_1 , y_1$), and likewise for all possible values for point $2$. And you calculate the Euclidean distance between those points. This must be normalized by the area of the sample space (1).

Done!

Answer 2

The quadruple integral will cycle though every possible pair of $(x_1,y_1),(x_2,y_2)$ in the unit square.

$\sqrt {(x_2-x_1)^2 - (y_2-y_1)^2}$ will give you the distance between those points.

We add them all up and divide by the area of the unit square (which is $1$)