The area (in sq. units) bounded by the curve f(x) = \ cases e^ -...

$∵$ Graph of $f \left(x\right)$ is symmetric about $x$-axis, hence,
Area $= 2\mathop{∫}$_{0}^{\iinfty}\left|f\left(x\right)\right|dx
= 2\left[\dfrac{1}{2} \times \left(1 - \dfrac{1}{e}\right)\left(1 - \dfrac{1}{e}\right) + \dfrac{1}{2} \times \dfrac{1}{e} \times \dfrac{1}{e} +$\mathop{∫}_{1}^{\infty}$ e^{- x}dx\right]$<br />$= 2\left[$\dfrac{1}{2} \left[1 - \dfrac{2}{e} + \dfrac{2}{e^{2}}\right] + {\left[- e^{- x}\right]}$_{1}^{\infty}\right]$<br />$=\left[1 - \dfrac{2}{e} + \dfrac{2}{e^{2}} + \dfrac{2}{e}\right] =1 + \dfrac{2}{e^{2}}& sq. units