The area of the region enclosed by the curves , and y-axis is: [2025]

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Solution

Q.
The area of the region enclosed by the curves $y = e^{x}$ , $y = | e^{x} - 1 |$ and y-axis is: [2025]

1 $\log_{e} 2$

2 $1 - \log_{e} 2$

3 $2 \log_{e} 2 - 1$

4 $1 + \log_{e} 2$

Ans.
(2)

Given: Region is bounded by curves $y = e^{x}$ and $y = | e^{x} - 1 |$ .

The graph is given by

When $e^{x} = | e^{x} - 1 |$

$\Rightarrow e^{x} = - e^{x} + 1$ $[∵ e^{x} - 1 < 0]$

$\Rightarrow 2 e^{x} = 1 \Rightarrow e^{x} = 1 / 2 \Rightarrow x = - \ln (2)$

Now, area is given by.

$A = \int_{- \ln (2)}^{0} [e^{x} - (1 - e^{x})] d x = \int_{- \ln (2)}^{0} (2 e^{x} - 1) d x$

$= {(2 e^{x} - x)}_{- \ln (2)}^{0} = 2 - 1 - \ln (2) = 1 - \ln (2)$ .

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