Question

The area bounded by the curves $y=lnx,y=ln\left|x\right|,y=\left|lnx\right|$ and $y=\left|ln\left|x\right|\right|$ , for $x\in \left(-1,1\right)$ is (in sq. units)

Answer 1

Answer: 4

The required shaded region is as shown in figure

As the graph is symmetric in all quadrants, we calculate area in one quadrant and multiply by $4$ .
Hence, required area $=4|{\int }_0^1\left(lnx\right).dx|$
$A=4\left|{\left[xlnx-x\right]}_0^1\right|$
$A=4\left|\left[\left(1ln1-1\right)-\underset{x\to 0^{+}}{\lim }\left(xlnx-x\right)\right]\right|$
$A=4\left|\left[\left(0-1\right)-\underset{x\to 0^{+}}{\lim }\frac{lnx}{\frac{1}{x}}-0\right]\right|=4\left|\left[-1-\underset{x\to 0^{+}}{\lim }\frac{\frac{1}{x}}{-\frac{1}{x^2}}\right]\right|$
$A=4\left|\left[-1+\underset{x\to 0^{+}}{\lim }x\right]\right|$
$A=4\left|-1\right|$
$A=4$ sq. units