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Area of the region bounded by the curve $y=e^x$ and lines $x=0$ and $y=e$ is
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2866 Upvotes
Verified Answer
The correct answers are:
$\int_1^e \ln (e+1-y) d y$
,
$e-\int_0^1 e^x d x$
,
$\int_1^e \ln y d y$
$\int_1^e \ln (e+1-y) d y$
,
$e-\int_0^1 e^x d x$
,
$\int_1^e \ln y d y$
Shaded area $=e-\left(\int_0^1 e^x d x\right)=1$
Also, $\quad \int_1^e \ln (e+1-y) d y$
$$
\begin{aligned}
& =\int_e^q \ln t(-d t) \\
& \text { [put } e+1-y=t \Rightarrow-d y=d t \text { ] } \\
& =\int_1^e \ln t d t=\int_1^e \ln y d y=1
\end{aligned}
$$
Also, $\quad \int_1^e \ln (e+1-y) d y$
$$
\begin{aligned}
& =\int_e^q \ln t(-d t) \\
& \text { [put } e+1-y=t \Rightarrow-d y=d t \text { ] } \\
& =\int_1^e \ln t d t=\int_1^e \ln y d y=1
\end{aligned}
$$
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