The area of the region given by [ (x,y):xy 8,1 y x^2 ] is :

Question

The area of the region given by $$\{ (x,y):xy \le 8,1 \le y \le {x^2}\} $$ is :

Accepted Answer

$\begin{aligned} & 	ext { Required area }=\int_1^2\left(x^2-1ight) d x+\int_2^8\left(\frac{8}{x}-1ight) d x \\ & =\left.\left(\frac{x^3}{3}-xight)ight|_1 ^2+\left.(8 \ln x-x)ight|_2 ^8 \\ & =\left[\left(\frac{8}{3}-2ight)-\left(\frac{1}{3}-1ight)ight]+[8 \ln 8-8-(8 \ln 2-2)] \\ & =\frac{4}{3}+8 \ln 4-6 \\ & =8 \ln 4-\frac{14}{3} \\ & =16 \ln 2-\frac{14}{3}\end{aligned}$

Solutions · 1