Q. The area of the region bounded by the curves $y = e^{x},\ y = \log_{e}x\ and\ lines\$ $x = 1,\ x = 2$ is
A) ${(e - 1)}^{2}$
B) $e^{2} - e + 1$
C) $e^{2} - e + 1 - 2\log_{e}2$
D) $e^{2} + e - 2\ \log_{e}2$
Explanation
(c)
Required area
$A = \int_{1}^{2}{\left( e^{x} - \log_{e}x \right)dx}$
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$= \lbrack e^{x}\rbrack_{1}^{2} - \left\lbrack x\ \log_{e}x - \int_{}^{}{1dx} \right\rbrack_{1}^{2}$
${= e}^{2} - e - \lbrack x\log_{e}x - x\rbrack_{1}^{2}\$
$= e^{2} - e - \left\lbrack 2\log_{e}2 - 2 - (0 - 1) \right\rbrack$
$= e^{2} - e - 2\log_{e}2 + 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.