Question

Evaluate the following integral if n is an integer and D is the region bounded by x^2+y^2=a^2 and x^2+y^2=b^2 where 0< a < b. ∫ ∫ D (1) / [(x^2+y^2)^(n/2)] dA. What happens to your answer in the limit a→0^+? Note: D should be at the bottom of the second interval, it is not part of the function.

Answer 1

$\iint_D\frac{dA}{(x^2+y^2)^{n/2}}$

Let's use the polar coordinates

$dA=(rd\theta)(dr)$

Where a<r<b

$\iint_D\frac{rd\theta dr}{r^{2n/2}}=\int\limits_{a}^{b} r^{1-n}dr\int\limits_{0}^{2\pi}d\theta=2\pi\left(\frac{r^{2-n}}{2-n}\right)_a^b=2\pi\frac{b^{2-n}-a^{2-n}}{2-n}$

If $a\to 0$, then

$2\pi\frac{b^{2-n}-a^{2-n}}{2-n}\to2\pi\frac{b^{2-n}}{2-n}$