The area of the region S = ( x , y ) : 3 x 2 ≤ 4 y ≤ 6 x + 24 is______.

Step-by-step solution

We have, S = ( x , y ) : 3 x 2 ≤ 4 y ≤ 6 x + 24 Then, y = 3 x 2 4 4 y = 6 x + 24 Solving both equations, we get - 2 , 3 and 4 , 12 . Required area A = ∫ − 2 4   6 x + 24 4 − 3 x 2 4 d x ⇒ A = 1 4 ∫ − 2 4   6 x + 24 − 3 x 2 d x ⇒ A = 3 4 x 2 + 6 x − x 3 4 − 2 4 ⇒ A = 12 + 24 − 16 - 3 + 12 - 2 ⇒ A = 27   sq . units