The area of the region A = x , y : cos x - sin x ≤ y ≤ sin x , 0 ≤ x ≤ π 2

Step-by-step solution

Given, cos   x - sin   x ≤ y ≤ sin   x ,   0 ⩽ x ⩽ π 2 For intersection point: cos x - sin x = sin x ⇒ tan x = 1 2 Let ψ = tan - 1 1 2 , so tan   ψ = 1 2 sin   ψ = 1 5 cos   ψ = 2 5 Required area is = ∫ ψ π / 2 sin x - cos x - sin x d x = ∫ ψ π / 4 sin x - cos x - sin x d x + ∫ π / 4 π / 2 sin x + cos x - sin x d x = ∫ y π / 4 2 sin x - cos x d x + ∫ π / 4