The domain here is the rectangle $[-1,1]\times [-1,1]$ and a region is cut out given by the semi circle $y=-\sqrt{1-x^2}$.
If the domain we are integrating over is then defined as $\Omega$ is $\int \int _{\Omega }f(x,y) \ dA$ given by $$\int_{-1}^{1} dx \int_{-1}^{1} f(x,y) \ dy -\int_{-1}^{1} dx\int_{-\sqrt{1-x^2}}^{0} f(x,y) \ dy $$ or $$\int_{-1}^{1} dx \int_{-1}^{1} f(x,y) \ dy +\int_{-1}^{1} dx\int_{-\sqrt{1-x^2}}^{0} f(x,y) \ dy $$?
[The domain here is supposed to be the black rectangle with the blue semi circle cut out.]