Question

the area bounded by the curves y = log_e x , y = log_e |x|_1, y = | log_e x| and y = |log_e|x|| is​

Answer 1

firstly draw them and combine on a single plane.

( refer to Attachment)

now, apply L'hôpital's rule,

$\displaystyle\sf \lim_{x\to 0} ( x \ log\ x - x)$

$\displaystyle\sf = \lim_{x\to 0} \left( \dfrac{log \ x-1}{\dfrac{1}{x}} \right) \:\; \left( \dfrac{\infty}{\infty} \right)$

$\displaystyle\sf = \lim_{x\to 0} \dfrac{1/x}{-1/x^2} = \lim_{x\to 0} (-x) = 0$

area $\displaystyle\sf = 4 \left| \int\limits_0^1 ( log_e \ x) dx \right|$

$\displaystyle\sf = 4 \left| \Bigg[ x \ log \ x-x \Bigg]^1_0 \right|$

$\displaystyle\sf = 4 \left| 1 - \lim_{x\to 0} ( x \ log \ x-x) \right|$

$\sf = 4 |1-0|$

$\sf = 4|1|$

we know that

$\large\sf { |x| = \begin{cases} \sf x \;\text{if}\; x \geq 0 \\\\ \sf -x\; \text{if}\; x < 0 \end{cases} }$

and here $\sf x > 0$

$\sf = 4 \times 1$

$\sf = 4 \ sq. \ units$