Question

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

Answer 1

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 $=4\left|\right. \displaystyle \int _{0}^{1} \left(l n x\right) . d x \left|\right.$
$A=4\left|\left[x ln x - x\right]_{0}^{1}\right|$
$A=4\left|\left[\left(1 ln 1 - 1\right) - \underset{x \rightarrow 0^{+}}{lim} \left(x ln x - x\right)\right]\right|$
$A=4\left|\left[\left(0 - 1\right) - \underset{x \rightarrow 0^{+}}{lim} \frac{ln x}{\frac{1}{x}} - 0\right]\right|=4\left|\left[- 1 - \underset{x \rightarrow 0^{+}}{lim} \frac{\frac{1}{x}}{- \frac{1}{x^{2}}}\right]\right|$
$A=4\left|\left[- 1 + \underset{x \rightarrow 0^{+}}{lim} x\right]\right|$
$A=4\left|- 1\right|$
$A=4$ sq. units