Question

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

• A. $16 \log _e 2-\frac{14}{3}$
• B. $8 \log _e 2-\frac{13}{3}$
• C. $8 \log _e 2+\frac{7}{6}$
• D. $16 \log _e 2+\frac{7}{3}$

Answer 1

Answer: (A)

Area $=\int\limits_1^2\left(x^2-1\right) d x+\int\limits_2^8\left(\frac{8}{x}-1\right) d x$
$ =\left(\frac{x^3}{3}\right)_1^2+8(\ell n x)_2^8-(x)_1^8 $
$ =\frac{7}{3}+8(2 \ell n 2)-7 $
$ =16 \ell n 2-\frac{14}{3}$