The area of the region given by \ (x, y): xy ≤ 8, 1 ≤ y ≤ x ^ 2 \...

Given,

$xy \le 8 \text{ and }$ 1 \le y \le {x}^{2}$</p><p>On solving above curve we get intersecting point as \left(2, 4\right)$

Now on plotting the diagram of the given curves we get,

Required area

$A = {∫}_{1}^{2}\left({x}^{2} - 1\right)dx + {∫}_{2}^{8}\left(\dfrac{8}{x} - 1\right)dx$

$⇒ A = {∫}_{1}^{2}{x}^{2}dx + {∫}_{2}^{8}\dfrac{8}{x}dx - {∫}_{1}^{8}dx$

$⇒ A = {\left[\dfrac{{x}^{3}}{3}\right]}_{1}^{2} + 8{\left[\ln x\right]}_{2}^{8} - {\left[x\right]}_{1}^{8}$

$⇒ A = 8\ln 4 - \dfrac{14}{3}$

$⇒ A = 16\ln 2 - \dfrac{14}{3}$ sq. units