The area (in sq. units) of the region A = x , y : x 2 ≤ y ≤ x + 2 is

Step-by-step solution

The two given curves are y = x + 2 and y = x 2 To find their points of intersection, we equate the value of y to get x 2 = x + 2 ⇒ x 2 - x - 2 = 0 ⇒ x - 2 x + 1 = 0 ⇒ x = 2 or x = - 1 . The graphs of the curves are given below. Required area is the area of the shaded portion. Area = ∫ - 1 2 y l i n e - y p a r a b o l a d x = ∫ - 1 2 x + 2 - x 2 d x Using ∫ x n d x = x n + 1 n + 1 = x 2 2 +   2 x - x 3 3 - 1 2 = 2 + 4 - 8 3 - 1 2 - 2 + 1 3 = 6 - 8 3 - 1 2 +   2 - 1 3 =