If the area of the bounded region R = x , y : max 0 , log e x ≤ y ≤ 2 x , 1 2 ≤ x ≤ 2 is, α log e 2 - 1 + β log e 2 + γ then the value of α + β - 2 γ 2 is equal to:

Step-by-step solution

R = x , y : max 0 , log e x ≤ y ≤ 2 x , 1 2 ≤ x ≤ 2 So, the area of the region is ∫ 1 2 2 2 x d x - ∫ 1 2 ℓ n x d x = 2 x ln 2 1 / 2 2 - x ln x - x 1 2 = 2 2 - 2 1 / 2 log e 2 - 2 ln 2 - 1 = 2 2 - 2 log e 2 - 2 ln 2 + 1 ∴   α = 2 2 - 2 , β = - 2 , γ = 1 So, α + β - 2 γ 2 = 2 2 - 2 - 2 - 2 2 = - 2 2 = 2