I understand the principle of substituting variables in a double integral and using the Jacobian to multiply by the correct factor, but if I have a double integral in x and y, when is it justified to be able to interchange the x and y terms? For example in the following integral: $$\int \int \frac{1+2x^2}{1+x^4+6x^2y^2+y^4} -\frac{1+y^2}{2+x^4+y^4}dxdy$$
integrated over $x^2+y^2\leq R^2$ for some positive real $R$. Can I interchange the x and y terms without changing the value of the integral? I.e. is this integral equal to the following integral:$$\int \int \frac{1+2y^2}{1+y^4+6x^2y^2+x^4} -\frac{1+x^2}{2+x^4+y^4}dxdy$$ (x and y have been interchanged). If yes is it because the region where it is integrated in is circular thus independent of x and y? I am having a tough time understanding when we can do this and when we can't.