Switching the order of integration for a general, infinite region?

Question

This is probably a stupid question, but if you had a non-rectangular, infinite region R, e.g. $$a \leq x \leq \infty$$ $$f(x) \leq y \leq g(x)$$

is it even possible to switch the order of integration when evaluating the double integral: $$\iint_{R} f(x,y) dA$$

I don't think it is, but I wanted a second opinion.

Thanks!

Answer 1

If $f \geq 0$ we can take the order of integration any way we please, so that $\iint_R f(x,y) dA = \int dx \int dy 1_R(x,y) f(x,y) = \int dy \int dx 1_R(x,y) f(x,y)$, where $1_R$ is the indicator function on $R$, that is $1$ if $(x,y) \in R$ and $0$ otherwise.

If $\iint_R |f(x,y)|dA < \infty$, then you can also apply the above formula.

These are results of the Fubini-Tonelli Theorems.