The area of the region given by { ( x , y ) : x y 8 , 1 y x 2 } is [2023]

Question

The area of the region given by ${(x, y) : x y \leq 8, 1 \leq y \leq x^{2}}$ is [2023]

Smart Achievers · Accepted Answer

(1)

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

$= {(\frac{x^{3}}{3} - x)}_{1}^{2} + 8 {(\log_{e} x)}_{2}^{8} - {(x)}_{2}^{8}$

$= \frac{4}{3} + 8 (2 \log_{e} 2) - 6 = 16 \log_{e} 2 - \frac{14}{3}$

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