If the area of the region $\left\{(x, y):-1 \leq x \leq 1,0 \leq y \leq a+\mathrm{e}^{|x|}-\mathrm{e}^{-x},…
If the area of the region $\left\{(x, y):-1 \leq x \leq 1,0 \leq y \leq a+\mathrm{e}^{|x|}-\mathrm{e}^{-x}, \mathrm{a} \gt 0\right\}$ is $\frac{\mathrm{e}^2+8 \mathrm{e}+1}{\mathrm{e}}$, then the value of $a$ is :
8
7
5
6
Solution
$\begin{aligned} & y \in\left[0, a+e^{|x|}-e^{-x}\right] \\ & \text { (i) If } x \geq 0 \Rightarrow y \in\left(0, a+e^x-\frac{1}{e^x}\right) \\ & \text { if } x < 0 \Rightarrow y \in\left(0, a+e^{-x}-e^{-x}\right) \\ & \Rightarrow y \in(0, a)\end{aligned}$ $\begin{aligned} & \text { Area }=(a)+\int_0^1\left(a+e^x-e^{-x}\right) d x=\frac{e^2+8 e+1}{e} \\ & =a+\left.\left(a x+e^x+e^{-x}\right)\right|_0 ^1=e+8+\frac{1}{e} \\ & =a+\left(a+e+\frac{1}{e}-2\right)=e+\frac{1}{e}+8 \\ & \Rightarrow 2 a-2=8 \Rightarrow a=5\end{aligned}$
