If the area of the region S = ( x , y ) : 2 y - y 2 ≤ x 2 ≤ 2 y , x ≥ y is equal to n + 2 n + 1 - π n - 1 , then the natural number n is equal to _ _ _ _ _ _ _

Step-by-step solution

Given, S = ( x , y ) : 2 y - y 2 ≤ x 2 ≤ 2 y , x ≥ y , Now plotting the diagram, we get Now from above diagram, given region are, 2 y - y 2 ≤ x 2 , x 2 ≤ 2 y and x ≥ y Now the intersecting point of x 2 + y 2 - 2 y = 0   &   x = y will be 1 , 1 and x 2 = 2 y   &   x = y will be 2 , 2 So, required area will be, A = ∫ 0 1 1 + 1 - x 2 - x 2 2 + ∫ 1 2 x - x 2 2 d x ⇒ A = x - x 2 1 - x 2 - 1 2 · sin - 1 x - x 3 6 0 1 + x 2 2 - x 3 6 1