The area of the region { ( x , y ) : | x – y | y 4 x } is [2025]

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Solution

Q.
The area of the region ${(x, y) : | x - y | \leq y \leq 4 \sqrt{x}}$ is [2025]

1 512

2 $\frac{2048}{3}$

3 $\frac{1024}{3}$

4 $\frac{512}{3}$

Ans.
(3)

We have, $| x - y |$ $\leq y \leq 4 \sqrt{x}$

Now, $y =$ $| x - y |$ $\Rightarrow y^{2} = {(x - y)}^{2}$

$\Rightarrow y = \frac{x}{2} and x = 0$

Required Area $= \int_{0}^{64} (4 \sqrt{x} - \frac{x}{2}) d x$

$= {[\frac{4 x^{3 / 2}}{3 / 2} - \frac{x^{2}}{4}]}_{0}^{64} = \frac{8}{3} \cdot 8^{3} - \frac{64^{2}}{4} = 64^{2} (\frac{1}{12}) = \frac{1024}{3}$ .

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