Answer
\[A=1\]
Work Step by Step
\[\begin{align}
& \text{From the graph} \\
& R=\left\{ \left( x,y \right):0\le y\le {{e}^{x}},\text{ }0\le x\le \ln 2\text{ } \right\} \\
& \text{Then,} \\
& A=\int_{0}^{\ln 2}{\int_{0}^{{{e}^{x}}}{dy}dx} \\
& \text{Integrate} \\
& A=\int_{0}^{\ln 2}{\left[ y \right]_{0}^{{{e}^{x}}}dx} \\
& A=\int_{0}^{\ln }{\left( {{e}^{x}}-0 \right)dx} \\
& A=\int_{0}^{\ln 2}{{{e}^{x}}dx} \\
& A=\left[ {{e}^{x}} \right]_{0}^{\ln 2} \\
& A={{e}^{\ln 2}}-{{e}^{0}} \\
& A=2-1 \\
& A=1 \\
\end{align}\]
