Let for x ∈ R , f x = x + x 2 and g x = x , x 2 , x < 0 x ≥ 0 . Then area bounded by the curve y = f o g x and the lines y = 0 , 2 y - x = 15 is equal to _____ .

Step-by-step solution

We have, f x = x + x 2 = x ; x ≥ 0 0 ;       x < 0 g x = x 2 ; x ≥ 0 x ;       x < 0 So, f o g x = f g x = g x ; g x ≥ 0 0 ; g x < 0 ⇒ f o g x = x 2 ; x ≥ 0 0 ;       x < 0 And, 2 y - x = 15 Solving 2 y - x = 15 and y = x 2 , we get 2 x 2 - x - 15 = 0 ⇒ x = 1 ± 11 4 ⇒ x = 3 ,   - 5 2 Required area A = ∫ 0 3 x + 15 2 - x 2 dx + 1 2 × 15 2 × 15 ⇒ A = x 2 4 + 15 2 x - x 3 3 0 3