Question

The area of the region bounded by the curve $y=e^x$ and lines $x=0$ and $y=e$ is

• A. $e-1$
• B. $\underset{1}{\overset{e}{\int }}\ln (e+1-y)dy$
• C. $e-\underset{0}{\overset{1}{\int }}{e}^{x}dx$
• D. $\underset{1}{\overset{e}{\int }}\ln ydy$

Answer 1

Answer: (B), (C), (D)

$\underset{1}{\overset{e}{\int }}\ln ydy$

The area bounded by the curve $y=e^x$ and lines $x=0$ and $y=e$ is as shown in the graph.

Required area$=($area of rectangle formed by lines $x=0,x=1,y=0,y=e)-\left(\underset{0}{\overset{1}{\int }}{e}^{x}dx\right)$
$=e-\underset{0}{\overset{1}{\int }}{e}^{x}dx=1$
Also required area
$=\underset{0}{\overset{e}{\int }}xdy($where $e^x=y\Rightarrow x=\ln y)$
$=\underset{1}{\overset{e}{\int }}\ln ydy$
$=\underset{1}{\overset{e}{\int }}\ln (e+1-y)dy$
$[\because \underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f(a+b-x)dx]$