Consider a region R = x , y ∈ R 2 : x 2 ≤ y ≤ 2 x . If a line y = α divides the area of region R into two equal parts, then which of the following is true ?

Step-by-step solution

∫ 0 4 y - y 2 d y = 2 ∫ 0 α y - y 2 d y 2 y 3 / 2 3 - y 2 2 0 4 = 2 2 y 3 / 2 3 - y 2 4 0 α ⇒ 16 3 - 4 = 2 2 3 α 3 / 2 - α 2 4 ⇒    4 3 = 2 2 3 α 3 / 2 - α 2 4 ⇒    2 3 = 8 α 3 / 2 - 3 α 2 12 ⇒    8 α 3 / 2 - 3 α 2 = 8 ⇒ 3 α 2 - 8 α 3 / 2 + 8 = 0