Question

The area of the region \left\\{ (x, y) : y^2 \leq 4x, \, x<4, \, \frac{xy(x - 1)(x - 2)}{(x - 3)(x - 4)}>0, \, x \neq 3 \right\\}is

• A. $\frac{16}{3}$
• B. $\frac{64}{3}$
• C. $\frac{8}{3}$
• D. $\frac{32}{3}$

Answer 1

Answer: (D)

$\frac{32}{3}$

Answer 2

$y^2 \leq 4x, \quad x < 4$

$\frac{xy(x-1)(x-2)}{(x-3)(x-4)} > 0$

Case - I: $y > 0$

$\frac{x(x-1)(x-2)}{(x-3)(x-4)} > 0, \quad x \in (0,1) \cup (2,3)$

Case - II: $y < 0$

$\frac{x(x-1)(x-2)}{(x-3)(x-4)} < 0, \quad x \in (1,2) \cup (3,4)$

Area:

$\text{Area} = 2 \int_{0}^{4} \sqrt{x} \, dx$

$= 2 \cdot \frac{2}{3} \left[ x^{3/2} \right]_{0}^{4} = \frac{32}{3}$