Answer
$\ln (2)$
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
Thus, the area of the region is:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{0}^{\ln 2} (2 e^{-x}+1-e^x) \ dx \\=[-2e^{-x}+x-e^x]_0^{\ln 2}\\=[-2e^{-\ln 2}+\ln 2-e^{\ln 2}-(-2+0-1) \\=\ln (2)$
