GATE 2021 ST – Question 21
Let $I=4\int_{1/\sqrt2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}\,dy\,dx$. Find $e^{I+\pi}$,to two decimals.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 33.50 to 34.50
Explanation
Use polar coordinates. The region has 0≤θ≤π/4 and 1≤r≤sec θ. The Jacobian cancels the integrand 1/r,so I=4∫₀^(π/4)(sec θ−1)dθ=4ln(√2+1)−π. Thus e^(I+π)=(1+√2)⁴=17+12√2≈**33.97**.