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 [2024]

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Solution

Q.
The area of the region

${(x, y) : y^{2} \leq 4 x, x < 4, \frac{x y (x - 1) (x - 2)}{(x - 3) (x - 4)} > 0, x \neq 3}$ is [2024]

1 $\frac{32}{3}$

2 $\frac{16}{3}$

3 $\frac{8}{3}$

4 $\frac{64}{3}$

Ans.
(1)

The area of region $x y \frac{(x - 1) (x - 2)}{(x - 3) (x - 4)}, y^{2} - 4 x \leq 0$

$\Rightarrow y x (x - 1) (x - 2) (x - 3) (x - 4) < 0$

Case 1: $y > 0$

So, $x (x - 1) (x - 2) (x - 3) (x - 4) < 0$

$\Rightarrow x \in (0, 1) \cup (2, 3)$

Case 2: $y < 0$

So, $x (x - 1) (x - 2) (x - 3) (x - 4) > 0 \Rightarrow x \in (1, 2) \cup (3, 4)$

$Area = \int_{0}^{4} 2 \sqrt{x} d x = \frac{2}{3} \times 16 = \frac{32}{3}$

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