Question

Let \(D=\{(x,y)\in\mathbb{R}^2:0<x^2+y^2\leq1\}\). For \(\alpha\geq0\), consider the integral
\[ I_\alpha=\iint_D \frac{1}{(x^2+y^2)^\alpha}\,dx\,dy. \]
Let
\[ N_0=\sup\{\alpha\geq0:I_\alpha<\infty\}. \]
Then the value of \(N_0\) equals ______ (answer in integer).

Hint:

Switch to polar coordinates and check when \(\int_0^1 r^{1-2\alpha}dr\) converges near \(r=0\).

Answer 1

Correct Answer: 1

Step 1: Polar form.
$x=r\cos\theta$, $y=r\sin\theta$; punctured disc becomes $0<r\leq1$, full angle sweep.

Step 2: Rewrite integral.
\[ I_\alpha=2\pi\int_0^1 r^{1-2\alpha}\,dr \]

Step 3: p-test near zero.
$\int_0^1 r^p\,dr$ finite iff $p>-1$, here $p=1-2\alpha$.

Step 4: Bound on alpha.
\[ 1-2\alpha>-1 \Rightarrow \alpha<1 \]

Step 5: Supremum.
Set is $[0,1)$, supremum $1$.
\[ \boxed{N_0=1} \]