Question

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

• A. $\frac{1}{2}+\ln 2\text{sq. units}$
• B. $3+\ln 2\text{sq. units}$
• C. $\frac{31}{4}\text{sq. units}$
• D. $1+2\ln 2\text{sq. units}$

Answer 1

The correct option is B $3+\ln 2\text{sq. units}$

$max(|x|,|y|)\le 1$
$\Rightarrow |x|\le 1$ and $|y|\le 1$
which represent a square bounded by $x=\pm 1$ and $y=\pm 1$


The required area is the shaded area.
Now, unshaded area is given by
$A=2\underset{1/2}{\overset{1}{\int }}(1-\frac{1}{2x})dx$
$=2{(x-\frac{1}{2}\ln x)}_{1/2}^1$
$=2\left[\right(1-0)-(\frac{1}{2}-\frac{1}{2}\ln \frac{1}{2})]$
$=1-\ln 2\text{sq. units}$

$\therefore$ Required(shaded) area $=4-(1-\ln 2)=3+\ln 2\text{sq. units}$