Question

The area of region enclosed by the curves \(y = e^x, y=|e^x-1|\) and y – axis is (in sq. units) 

(1) 1 

(2) 1 – ln2 

(3) 1 + ln2 

(4) ln2

Answer 1

Correct option is (2) 1 – ln2  

\(\Rightarrow e^x = \frac{1}{2}\)    

\(\Rightarrow x = \text{In} \frac{1}{2}\)    

\(\Rightarrow \int_{\ln \left(\frac{1}{2}\right)}^0\left[e^x-\left(1-e^x\right)\right] d x\)  

\(=\int_{\ln 2}^0\left(2 e^x-1\right) d x=2 e^x-\left.x\right|_{-\ln 2} ^0\)   

\(=2-(1+\ln 2)\)

\(=1-\ln 2\)