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Q.
The area bounded by the curves $y = xe^x$, $y = xe^{-x}$ and the line $x = 1$ is
Application of Integrals
Solution:
Given curves are $y = xe^x$ and $y = xe^{-x}$
Line $x = 1$ meets the curves at $A\left(1, e\right)$ and $B \left(1, \frac{1}{e}\right)$.
Both the curves pass through origin.
$\therefore $ Required area $= \int\limits^{1}_{0}\left(xe^{x}-xe^{-x}\right)dx$
$= \int\limits^{1}_{0}x\left(e^{x}-e^{-x}\right)dx$
=$x\left(e^{x}-e^{-x}\right)|^{1}_{0}-\int\limits^{1}_{0}\left(e^{x}-e^{-x}\right)dx$
$=\left(e+\frac{1}{e}\right)-e^{x} \left|^{1}_{0} + e^{-x}\right|^{1}_{0}$
$= e +\frac{1}{e} - e + 1+\frac{1}{e} - 1 = \frac{2}{e}$
