It's easy to understand that when calculating integrals of distribution functions of one variable, we can get the following equation, which is corresponding to the calculation of differentials like $\mathrm{d}F(x) = f(x)\,\mathrm{d}x$. $$\int g(x)\,\mathrm{d}F(x) = \int g(x)f(x)\,\mathrm{d}x.$$
But, when it comes two dimensions, I don't know how to simplify or explain it via calculus. Is the following equation right? And how to explain it via differentials? $$\int g(x,y)\,\mathrm{d}F(x,y) = \iint g(x,y)f(x,y)\,\mathrm{d}x\,\mathrm{d}y.$$
So the answer is quite easy. IN your second equation, you first should integrate function by x variable and then by y variable. $$\int g(x,y)\,\mathrm{d}F(x,y) = \int[\int g(x,y)f(x,y)\,\mathrm{d}x\,]\mathrm{d}y.$$ So you will start like this