Can anybody solve this thing? I really need it as a part of my PHD. I already tried several professors that just didn't have a clue......
I would first encapsulate it:
$\displaystyle \int_y\int_x g(x,y)dxdy$
and start off with a simple expression for $\displaystyle g(x,y)$ just to get the algebra of analysis correct.
I'm unclear about what you mean by your notation: For the inner integral, I assume that means, integrate with respect to x and replace all occurrences of x by 'x'. Same with the y. Wouldn't that just be antiderivatives? Would need constants of integrations in that case and not sure how to handle it symbolically that way. Personally, I'd do a brute-force numerical analysis on the function:
$\displaystyle f(x,y;x_0,y_0)=\int_{y_0}^y\int_{x_0}^x g(u,v)dudv$ where $\displaystyle g(u,v)$ is your function.
You can do that pretty easily in Mathematica and even create an animation file showing how the function $\displaystyle f(x,y)$ changes as you vary the initial points $\displaystyle x_0,y_0$. Might be fun.