The area of the region x , y : x y ≤ 8 , 1 ≤ y ≤ x 2 is

Step-by-step solution

To draw the inequality, let us draw the equation x y = 8 and y = 1 and y = x 2 For point of intersection (i) x y = 8 and y = 1 ⇒ A ( 8,1 ) (ii) x y = 8 and y = x 2 ⇒ x 3 = 8 ⇒ x = 2 ⇒ B ( 2 , 4 ) (iii) y = x 2 and y = 1 ⇒ C 1 , 1 and D 1 , 1 Now x y ≤ 8 ⇒ region which contains origin y ≥ 1 ⇒ region above line y = 1 y ≤ x 2 ⇒ region outside the parabola Now required area Method I: Using x-axis: A = ∫ 1 2 x 2 - 1 ⋅ d x + ∫ 2 8 8 x - 1 ⋅ d x A = x 3 3 - x 1 2 + 8 l n x - x 2 8 A = - 14 3 + 16 l n 2 Method II: Using y-