If the area of the region (x, y):-1 x 1,0 y a+e^|x|-e^-x, a0 is e^2+8 e+1e, then the valu…

Question

If the area of the region $\left\{(x, y):-1 \leq x \leq 1,0 \leq y \leq \mathrm{a}+\mathrm{e}^{|x|}-\mathrm{e}^{-x}, \mathrm{a}>0\right\}$ is $\frac{\mathrm{e}^2+8 \mathrm{e}+1}{\mathrm{e}}$, then the value of $a$ is :

Accepted Answer

$$\begin{aligned}
& \text { required area is } a+\int_\limits0^1\left(a+e^x-e^{-x}\right) d x \
& a+\left[a+e^x+e^{-x}\right]_0^1 \
& 2 a+e-1+e^{-1}-1=e+8+\frac{1}{e} \
& 2 a=10 \Rightarrow a=5
\end{aligned}$$

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