I'm asked to compute: $$ \int_E \frac{1}{x^ay^b}dxdy \qquad E={x>0,y>0,xy \geq} 1$$ but I'm really finding difficulties in doing it as the domain is unlimited. I can pretty much solve this kind of integrals when the domain is limited and there is a problem in one or more than one points, but how can I compute this in an efficient way? I am also a little bit concerned about polar coordinates: I think that it could be a good way to do it, but people here, on other questions, told me that I should avoid doing it to prevent other issues. Can you please help me understanding how to proceed in problems like this one? Thanks in advance.
EDIT
We notice that we can express the domain as $E=\{x>0, y>\frac{1}{x}\}$ so we have:
$$\int_E \frac{1}{x^ay^b}dxdy= \int_0^{+\infty}\left(\int_{\frac{1}{x}}^{+\infty}\frac{1}{x^ay^b}dy\right)dx=\int_0^{+\infty}\frac{1}{x^a}\left[ \frac{1}{(1-b)y^{b-1}} \right]_{\frac{1}{x}}^{\infty}dx$$
Let's suppose $b>1$ (otherwise the integral diverges): $$\frac{-1}{b-1}\int_0^{+\infty} \frac{1}{x^a}x^{b-1}dx=\frac{-1}{b-1}\int_0^{+\infty} \frac{1}{x^{a-b+1}}dx$$ But now we have something that diverges for all values of $a,b$ as it has both problems in $0$ and at $\infty$.
Is this correct?
Because convergence of the double integral $$\int_E \frac{dxdy}{x^ay^b}$$ would imply convergence of both the iterated integrals $$\int_0^\infty \frac 1{x^a}\int_{1/x}^\infty \frac{dy}{y^b}dx$$ and $$\int_0^\infty \frac 1{y^b}\int_{1/y}^\infty \frac{dx}{x^a}dy,$$ the fact that $\int_0^\infty \frac 1{x^a}\int_{1/x}^\infty \frac{dy}{y^b}dx$ diverges to $\infty$ is enough to prove that $\int_E \frac{dxdy}{x^ay^b}$ does not converge.
(It is not enough to prove it diverges to $\infty$ as well. It could potentially diverge in other ways, but it cannot converge while one of the iterated integrals diverges.)
So your calculation is enough to show the integral does not converge.