The area of the region R = { ( x , y ) : x y 8 , 1 y x 2 , x 0 } is [2026]

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Solution

Q.
The area of the region $R = {(x, y) : x y \leq 8, 1 \leq y \leq x^{2}, x \geq 0}$ is [2026]

1 $\frac{1}{3} (49 \log_{e} (2) - 15)$

2 $\frac{2}{3} (20 \log_{e} (2) + 9)$

3 $\frac{2}{3} (24 \log_{e} (2) - 7)$

4 $\frac{1}{3} (40 \log_{e} (2) + 27)$

Ans.
(3)

$A = \int_{1}^{2} (x^{2} - 1) d x + \int_{2}^{8} (\frac{8}{x} - 1) d x$

$A = 8 \log_{e} 4 - \frac{14}{3} = 16 \log_{e} 2 - \frac{14}{3}$

$= \frac{2}{3} (24 \log_{e} 2 - 7)$

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