I want to decide on which of the following sets $A \subseteq \mathbb{R}^2$ the function $f(x, y) = e^{-x y} sin(x)$ is Lebesgue-integrable:
$$(i) A = [0, R] \times [0, \infty], R > 0;\space\space\space\space (ii) A = [0, \infty)^2 $$
Now, I believe (although I'm note entirely sure) it's sufficient to show that (at least) one of the integrals $\int_{[0, \infty]} ( \int_{[0, R]} |f(x, y)| dx) dy$ or $\int_{[0, R]} ( \int_{[0, \infty]} |f(x, y)| dy) dx$ exists and is $< \infty$? (Which, I believe, is a consequence out of Tonelli and Fubini?)
So therefore, if we consider $\int_{[0, R]} ( \int_{[0, \infty]} |f(x, y)| dy) dx = \int_{[0, R]} ( \int_{[0, \infty]} |\frac{sin(x)}{e^{xy}}| dy) dx = \int_{[0, R]} \frac{|sin(x)|}{x} dx$ (according to Wolframalpha, will try to manually check this later on), which exists for all $R > 0$, can we already say that $f$ is Lebesgue-integrable over the set $A$ given in $(i)$?
If I suspect that $f$ isn't Lebesgue-integrable over the set $A$, given in (ii), could I show that by showing that neither of the two iterated integrals exist and are $< \infty$?
If we consider that $\int_{[0, \infty)} ( \int_{[0, \infty)} |\frac{sin(x)}{e^{xy}}| dy) dx = \int_{[0, \infty)} \frac{|sin(x)|}{x} dx$, which I believe doesn't exist, would I need to check the other iterated integral aswell, or is it (according to Fubini's theorem) already enough to disprove that one of the integrals is $< \infty$?
I'm still relatively confused about the existence of Lebesgue integrals in multiple dimensions, and how to check if they do exist. Is there an easier or nicer way to see whether or not these two integrals exist?