Evaluate the double integral
$$ I = \int\int_D \frac{1}{(x^2 + y^2)^{n/2}} dxdy .$$
where $n$ is an integer and $D$ is the region of the plane bounded by two circles centered on the origin and with radii $R_1, R_2$, where $0 < R_1 < R_2$.
Use suitable coordinate system to evaluate $I$, showing the details of the coordinate transformation and how the answer depends on n. Also, for which values of $n$ will the integral converge as $R_1 > 0$ from above?
My friend at university (who is a couple years below me) gave me this question and I haven't got a clue! Not seen one this hard in a long time. Anyone have any ideas? It's annoying me like crazy.
Best Answer
Us polar coordinates, and your integral becomes $$2\pi \int_{R_1}^{R_2} r^{1-n} d r.$$ This should be easy to do.