Suppose we are given $\int_0^1 \int_0^1 f(x,y) dx dy $ and it converges but $\int_0^1 f(x,y) dx$ nor $\int_0^1 f(x,y) dy$ can be done in closed form. Also we cannot use symmetry. In that case I do not know how to do the double integral because the fact that $\int_0^1 \int_0^1 f(x,y) dx dy = \int_0^1 \int_0^1 f(x,y) dy dx $ does not seem to help.
How does one proceed ?
Do we try to cut the unit square into pieces and sum over the integrals of the pieces ?
Do we use a theorem such as Green's theorem ?
Are there other ways to proceed if these all fail ?
I was thinking about a way to still use the order of integration as a trick :
$\int_0^1 \int_0^1 f(x,y) dx dy = \int_0^1 \int_0^1 \int_0^1 f_1(x,y,z) dx dy dz$
So that we have a triple integral and then use Fubini's theorem (changing the order of integration).
Or maybe even $\int_0^1 \int_0^1 f(x,y) dx dy = \int_0^1 \int_0^1 ... \int_0^1 f_1(x,y,z) dx dy ... d\alpha$.
However how do I know if this will be successfull or not ? And how do I know how many variables I need to add ?
If that also fails, does that mean the double integral cannot be done in closed form ??
