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: Understand the region D.
Unit disc with centre removed. Polar coordinates: \(0<r\leq1\), \(0\leq\theta<2\pi\).

Step 2: Convert to polar.
\[ I_\alpha=\int_0^{2\pi}\int_0^1 \frac{1}{r^{2\alpha}}\cdot r\,dr\,d\theta = 2\pi\int_0^1 r^{1-2\alpha}\,dr \]

Step 3: Convergence condition.
\(\int_0^1 r^{1-2\alpha}dr\) converges near \(r=0\) iff \(1-2\alpha>-1\), i.e. \(\alpha<1\).

Step 4: At \(\alpha=1\).
\(\int_0^1 r^{-1}dr\) diverges.

Step 5: Supremum.
Set of good \(\alpha\) is \([0,1)\), supremum \(=1\).

Final Answer: \[ \boxed{N_0=1} \]