Exam

NTA Abhyas

Year

2020

Subject

Mathematics

Chapter

Area Under Curves

NTA Abhyas 2020 · Mathematics · Area Under Curves

The area bounded by the curves y = x , y = |x| , y = | x| and y = |...

The area bounded by the curves y = \text\ln x \text{,} y = \text\ln \left|x\right| \text{,} y = \left|\text\ln x\right| and y = \left|\text\ln \left|x\right|\right|, for $x \in ( - 1, 1 )$ is

Step-by-step solution

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 $= 4| \mathop{∫}_{0}^{1} ( \ln x ) . dx |$

$A = 4| {[ xlnx - x ]}_{0}^{1} |$

$A = 4| [ ( 1\ln 1 - 1 ) - \lim_{x \to {0}^{+}} ( xlnx - x ) ] |$

$A = 4| [ ( 0 - 1 ) - \lim_{x \to {0}^{+}} \dfrac{\ln x}{\dfrac{1}{x}} - 0 ] | = 4| [ - 1 - \lim_{x \to {0}^{+}} \dfrac{\dfrac{1}{x}}{- \dfrac{1}{{x}^{2}}} ] |$

$A = 4| [ - 1 + \lim_{x \to {0}^{+}} x ] |$

$A = + 4| - 1 |$

$A = 4$