Problem

Part I Multiple Choice

3. Consider the integral

$$I_{p,r}:=\iint_D \frac{1}{(x^2+y^2)^{p/2}}\,dA,$$

where $D$ is the region bounded by two concentric circles centred at the origin with radii $r$ and $1$ respectively, $0<r<1$. Let $J_p:=\lim_{r\to 0^+}I_{p,r}$.

(a) $J_p$ is convergent if $p<1$.

(b) $J_p$ is convergent if $1<p<2$.

(c) $J_p$ is convergent if $2<p$.

Among the above three statements, how many of them are true?

A) Only one.

B) Only two.

C) All of them.

D) None of them.