Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 328: 89

$$ e^{2} +1.$$

First, we find the intersections between the two curves by putting $$ e^2=e^x\Longrightarrow x=2.$$ Now, the area is given by \begin{align*} Area\ &=\int_0^2e^{2}-e^{x}dx \\ &=( xe^{2}-e^{x})_0^2\\ &=2e^{2} -e^{2} -(0-1)\\ &=e^{2} +1. \end{align*}