How to calculate the average x-coordinate of a region?

Question

I have a region in the x-y plane. Now I randomly select a point inside or on the boundary of the region. What is the expected value of the x-coordinate of my point? I.e. If I perform this operation infinitely many times, what is the average of the x-coordinate?

Thanks for your help!

Answer 1

$E[x] = \frac{\int\limits_{\cal R} x\ dA}{\int\limits_{\cal R} dA}$ where ${\cal R}$ is the two-dimensional region (assumed uniform density) and $dA$ is a differential area, such as $dx\ dy$ or $2 \pi r\ dr\ d\theta$.