Definite Integration Question 223
Question: The area bounded by the curves $ y=\ln x $ , $ y=\ln |x| $ , $ y=|\ln x| $ and $ y=|\ln |x|| $ is
[AIEEE 2002]
Options:
A) 4 sq. unit
B) 6 sq. unit
C) 10 sq. unit
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
We know that $ \log x $ is defined for $ x>0 $ and $ \log |x| $ is defined for all $ x\in R-{0} $
Also $ |\log x|\ge 0 $ and $ |\log |x||\ge 0 $
$ \therefore $ Required area is symmetrical in all the four quadrants and is equal to $ =4\int_0^{1}{|\log x|dx=-4\int_0^{1}{\log xdx}} $ , ( $ \because $ In $ (0,1),\log x<0) $ = $ -4[x\log x-x]_0^{1}=-4(-1)=4 $ sq.unit, $ ( \because \underset{x\to 0}{\mathop{\lim }}x\log x=0 ) $ .
BETA
