Q80 / 120 · VITEEE 2014 · Mathematics

The area bounded by $y=x e^{|x|}$ and lines $|x|=1$, $y=0$ is

  • A
    4 sq units
  • B
    6 sq units
  • C
    $1 \mathrm{sq}$ unit
  • Correct answer
    D
    2 sq units

Solution

$\quad|\mathrm{x}|=1$ $\therefore \quad \mathrm{x}=\pm 1$ $\therefore \quad \mathrm{y}=\mathrm{xe}^{|\mathrm{x}|}=\left\{\begin{array}{l}\mathrm{x} \cdot \mathrm{e}^{-\mathrm{x}},-1 < \mathrm{x} < 0 \\ \mathrm{x} \cdot \mathrm{e}^{\mathrm{x}}, 0 < \mathrm{x} < 1\end{array}\right\}$ $\therefore$ Required area $=\left|\int_{-1}^{0} \mathrm{xe}^{-\mathrm{x}} \mathrm{dx}+\int_{0}^{1} \mathrm{xe}^{\mathrm{x}} \mathrm{dx}\right|$ $=\left|\left[-\mathrm{x} \cdot \mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}\right]_{-1}^{0}+\left\{\mathrm{x} \cdot \mathrm{e}^{\mathrm{x}}-\mathrm{e}^{\mathrm{x}}\right\}_{0}^{1}\right|$ $=|\{(0-1)-(1 \cdot \mathrm{e}-\mathrm{e})\}|+|\{(\mathrm{e}-\mathrm{e})-(0-1)\}|$ $=1+1=2 \mathrm{sq}$ units

Hence, the correct answer is option: 'D'.

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