The area of the region enclosed by the parabola y = 4 x − x 2 and 3 y = x − 4 2 is equal to

Step-by-step solution

Given: y = 4 x - x 2 and 3 y = x - 4 2 ⇒ 4 x - x 2 = x - 4 2 3 ⇒ 12 x - 3 x 2 = x 2 + 16 - 8 x ⇒ 4 x 2 - 20 x + 16 = 0 ⇒ x 2 - 5 x + 4 = 0 ⇒ x - 4 x - 1 = 0 ⇒ x = 4 , 1 ⇒ y = 0 , 3 So, the intersection points are 4 , 0 and 1 , 3 . Area = ∫ 1 4 4 x − x 2 − x − 4 2 3 d x Area = 4 x 2 2 − x 3 3 − x − 4 3 9 1 4 = 64 2 − 64 3 − 4 2 + 1 3 − 27 9 = 27 − 21 = 6