Question

The area of the region bounded by the curves $y=x{e}^x,y=x{e}^{-x}$ and the line $\left|x\right|=1,y=0$ is:

• A. $4$
• B. $3$
• C. $2$
• D. $1$

Answer 1

Answer: (C)

$2$

Since $\left|x\right|=1\therefore x=\pm 1$
$y=x{e}^{\left|x\right|}=\{\begin{array}{c}x{e}^{-x}-1<x<0\\ {xe}^{x}0\le x<1\end{array}$
Therefore $=\left|{\int }_{-1}^0{xe}^{-x}dx\right|+\left|{\int }_0^1{xe}^{x}dx\right|=\left|{[{-xe}^{-x}-e^{-x}]}_{-1}^0\right|+\left|{[{xe}^x-e^x]}_0^1\right|=2$