Question

The area bounded by $y=2-|2-x|,y=\frac{3}{\left|x\right|}$ is

• A. $\frac{5-4\ln 2}{3}$.sq.unit
• B. $\frac{2-\ln 3}{2}$.sq.unit
• C. $\frac{4-3\ln 3}{2}$.sq.unit
• D. none of these

Answer 1

The correct option is C $\frac{4-3\ln 3}{2}$.sq.unit

$y=2-|2-x|,y=\frac{3}{\left|x\right|}$
$y=\{\begin{array}{c}x;x\le 2\\ 4-x;x\ge 2\end{array}$ and
$y=\{\begin{array}{c}\frac{3}{x};x>0\\ -\frac{3}{x};x<0\end{array}$
Hence,Required Area$PQRSP=$area$PQRP+$area$PRSP$
$=\left|{\int }_{\sqrt{3}}^2(x-\frac{3}{x})dx\right|+\left|{\int }_2^3[(4-x)-\frac{3}{x}]dx\right|$
$={[\frac{x^2}{2}-3\ln x]}_{\sqrt{3}}^2+{[4x-\frac{x^2}{2}-3\ln x]}_2^3$
On simplification,we get
$=\frac{4-3\ln 3}{2}$.sq.unit