Definite Integration Question 292
Question: The area bounded by the curves $ y={\log _{e}}x $ and $ y={{({\log _{e}}x)}^{2}} $ is
[RPET 2000]
Options:
A) $ 3-e $
B) $ e-3 $
C) $ \frac{1}{2}(3-e) $
D) $ \frac{1}{2}(e-3) $
Show Answer
Answer:
Correct Answer: A
Solution:
Required area $ =\int_1^{e}{[\log x-{{(\log x)}^{2}}]}dx $
$ A=\int_1^{e}{\log xdx}-\int_1^{e}{{{(\log x)}^{2}}dx} $
$ =[x\log x-x] _1^{e}-[x{{(\log x)}^{2}}-2x\log x+2x] _1^{e} $
$ =[e-e-(-1)]-[e{{(1)}^{2}}-2e+2e-(2)] $
$ =(1)-(e-2) $
$ =3-e $ .
BETA
