If the area bounded by the curves x 2 + y 2 ≤ 4 , x + y ≥ 2 , and y ≥ 1 is 2 π K - K 2 - 1 2 sq. units, then the value of K is equal to

Step-by-step solution

The required area is A = ∫ 1 2 4 - y 2 d y - A r e a ∆ A P Q ⇒ A = y 4 - y 2 2 + 4 2 sin - 1 ⁡ y 2 1 2 - 1 2 × 1 × 1 ⇒ A = 0 + 2 sin - 1 ⁡ 1 - 3 2 - 2 sin - 1 ⁡ 1 2 - 1 2 = π - 3 2 - π 3 - 1 2 = 2 π 3 - 3 2 - 1 2 sq. units