Answer
$\dfrac{81}{32}$
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:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{0}^{\ln 4} [e^x-e^{-2x}] \ dx\\= [e^x+\dfrac{e^{-2x}}{2}]_{0}^{\ln 4} \\ =(e^{\ln 4}+\dfrac{e^{-2\ln 4}}{2})-(e^0+\dfrac{e^{0}}{2})\\=(4+\dfrac{1}{2} \times e^{\ln 16})-(1+\dfrac{1}{2}) \\=\dfrac{81}{32}$