This is probably an elementary result, but every time I need it I'm always confused. I also looked for the solution but I could not find it, so I think this will serve well as a reference for the future!
Here are the problems: consider $A_n = B(0,1) \subset \mathbb{R}^n$ (the unit ball centered at the origin) and $B^n = \mathbb{R}^n \setminus B(0,1)$.
$1)$ find all values of $p$ for which $$\int_{A_n}|x|^p\ dx < \infty;$$ $2)$ find all values of $q$ for which $$\int_{B_n}|x|^q\ dx < \infty.$$
I think (hope) that looking at how you characterize such values of $p$ and $q$ will help me to remember them! Thank you!
Try the $n = 1$ case first. That is very well-known (it is done wherever improper integrals are treated). Then you want to reduce the higher-dimensional case to the $n = 1$ case. There are several ways to do this -- a slick way would be to do a generalized version of polar coordinates where you integrate a constant function over all the spheres of radius $r$. But if you just set up the integral and write it as an iterated integral (by Fubini's Theorem), then it seems to me that nothing particularly clever is required to evaluate it: you can just use the one-variable Fundamental Theorem of Calculus $n$ times.
Have you tried to evaluate these integrals? How far did you get?