GATE 2026 ST – Question 61
On D={(x,y):0<x²+y²≤1}, let I_α=∬_D(x²+y²)^(−α)dxdy,α≥0. Find N₀=sup{α≥0:I_α<∞} (integer).
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Correct answer: 1 to 1
Explanation
Polar coordinates give I_α=2π∫₀¹r^(1−2α)dr. Integrability at zero requires 1−2α>−1, namely α<1. The convergence set is [0,1), whose supremum is **1**, although the integral diverges at that endpoint.