JEE Mains · Maths · STD 12 - 8. Application and integration
The area of the region enclosed by the curves \(y=\mathrm{e}^x, y=\left|\mathrm{e}^x-1\right|\) and \(y\)-axis is:
- A \(1-\log _{\mathrm{e}} 2\)
- B \(\log _{\mathrm{e}} 2\)
- C \(1+\log _e 2\)
- D \(2 \log _e 2-1\)
Answer & Solution
Correct Answer
(A) \(1-\log _{\mathrm{e}} 2\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & e^x=1-e^x \Rightarrow 2 e^x=1 \\ & \Rightarrow e^x=\frac{1}{2} \\ & \Rightarrow x=\ln \frac{1}{2} \\ & \int_{\ln (1 / 2)}^0\left[e^x-\left(1-e^x\right)\right] d x \\ & =\int_{\ln 2}^0\left(2 e^x-1\right) d x=2 e^x-\left.x\right|_{-\ln 2} ^0 \\ & =2-(1+\ln 2) \\…
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