Question

The area of the region of the plane bounded by $(|x|,|y|)\le 1\&xy\le \frac{1}{2}$ is :

• A. $4-\ln 2$
• B. $\frac{15}{4}$
• C. $2+2\ln 2$
• D. $3+\ln 2$

Answer 1

Answer: (D)

shaded area in $1^{\text{st }}$ quadrant is to be excluded from the area of the square of side 2
${=\underset{1/2}{\overset{1}{\int }}\left(1-\frac{1}{2x}\right)dx=x-\frac{1}{2}\ln x]}_{1/2}^1$
$=\frac{1}{2}+\frac{1}{2}\ln \frac{1}{2}=\frac{1}{2}-\frac{1}{2}\ln \frac{1}{2}$
$\therefore 2\text{ times the shaded area }=1-\ln 2$
$\therefore \text{ Required area }=2-(1-\ln 2)+2=3+\ln 2\Rightarrow D$