GATE 2023 MA – Question 30
If $\int_1^4\int_{-x}^x (x^2+y^2)^{-1}dy\,dx=\int_{-\pi/4}^{\pi/4}\int_{\sec\theta}^{\alpha\sec\theta}r^{-1}dr\,d\theta$, find α.
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Correct answer: 4 to 4
Explanation
Set x=r cosθ,y=r sinθ. The wedge −x≤y≤x has −π/4≤θ≤π/4. The bounds x=1,4 become r=secθ,4secθ, and the Jacobian changes 1/r² to 1/r. Thus **α=4**.