Let A = x , y ∈ ℝ 2 : y ≥ 0 , 2 x ≤ y ≤ 4 - x - 1 2 and B = x , y ∈ ℝ × ℝ : 0 ≤ y ≤ min 2 x , 4 - x - 1 2 . Then the ratio of the area of A to the area of B is

Step-by-step solution

Given, 2 x ≤ y ≤ 4 - x - 1 2   : y ≥ 0 ⇒ y 2 + x - 1 2 = 4 and y = 2 x Now plotting the diagram we get, Area of A : area of shaded region as shown in the above figure, Shaded portion = circular O A B C - Ar ∆ O A B = π 4 4 - 1 2 2 1 Hence, Area of A = π - 1 Now, plotting the diagram of 0 ≤ y ≤ min 2 x , 4 - x - 1 2 we get, Area of B : area of shaded region as shown in the above figure, Area of B = Ar ∆ A O B + Area of arc of circle A B C = 1 2 1 2 + &#9