The area (in sq. units) of the region bounded by the curves x 2 + 2 y - 1 = 0 , y 2 + 4 x - 4 = 0 and y 2 - 4 x - 4 = 0 in the upper half plane is _________.

Step-by-step solution

The given curves are x 2 + 2 y - 1 = 0 ,   y 2 + 4 x - 4 = 0 and y 2 - 4 x - 4 = 0 . Drawing their graphs, we get The required area is shown by the shaded part and since the curves y 2 + 4 x - 4 = 0 and y 2 - 4 x - 4 = 0 are symmetric about the y -axis, hence they have the same area. Hence, the required area = 2 ∫ 0 2 4 - y 2 4 d y - ∫ 0 1 1 - x 2 2 d x = 2 ∫ 0 2 1 4 4 - y 2 d y - ∫ 0 1 1 2 1 - x 2 d x = 2 1 4 4 y - y 3 3 0 2 - 1 2 x - x 3 3 0 1 = 2 1 4 8 - 8 3 - 0 - 1 2 1 - 1 3 - 0 = 2