Let's see: is just the definition of mass wrt a density function over a given region.

Polar coordinates: , and since the Jacobian of the change from cartesian to polar coordinates is then we get that the above original double integral turns into

. As you can see, the limits for , to cover the half upper unit circle, and r goes from 0 to 1 to cover all the interior of this half circle.

The rest is just simple integration.

Tonio