The area of the region x , y : y 2 ≤ 4 x , x < 4 , x y x - 1 x - 2 x - 3 x - 4 > 0 , x ≠ 3 is

Step-by-step solution

Given: y 2 ≤ 4 x , x < 4 , x y x - 1 x - 2 x - 3 x - 4 > 0 Case - I: y > 0 ⇒ x x - 1 x - 2 x - 3 x - 4 > 0 ⇒ x ∈ 0 , 1 ∪ 2 , 3 . . . i Case – II : y < 0 x x - 1 x - 2 x - 3 x - 4 < 0 ⇒ x ∈ 1 , 2 ∪ 3 , 4 . . . i i So, the required area is given by, A = ∫ 0 4 4 x d x Take the mirror image of strips in the first quadrant to find the area ⇒ A = 2 ∫ 0 4 x d x ⇒ A = 2 · 2 3 x 3 2 0 4 ⇒ A = 2 · 2 3 4 3 2 ⇒ A = 2 × 2 3 × 8 ⇒ A = 32 3