computing a tough improper integral Let $A = \{ (x,y) : x^2 + y^2 \geq 1 \} $. Let $f(x,y) = (x^2+y^2)^{-(2 + \epsilon)} $ for some fixed parameter $\epsilon > 0 $. How can I compute $\int_A f(x,y) $. I know it is an improper integral, but I have trouble trying to set up this integral. 

I am just expliciting Jonathan's suggestion in the comments. Writing in polar coordinates you get $$\iint\limits_{A} f(x,y) = \iint\limits_{A} (x^2+y^2)^{-(2+\varepsilon)} \, dA = \int_0^{2\pi} \hspace{-5pt} \int_1^{\infty} (r^2)^{-(2+\varepsilon)} r \, dr \, d \theta.$$ Expanding, simplifying and computing the $\theta$ integral gives $$2\pi \int_1^{\infty} r^{-(3+2\varepsilon)} \, dr.$$ It is simple to justify convergence and compute.
