Answer
\[\int_{0}^{2}{\int_{{{e}^{y}}}^{{{e}^{2}}}{f\left( x,y \right)}dxdy}\]
Work Step by Step
\[\begin{align}
& \int_{1}^{{{e}^{2}}}{\int_{0}^{\ln x}{f\left( x,y \right)}dydx} \\
& y=\ln x\to x={{e}^{y}} \\
& \text{Using the graph to switch the order of integration} \\
& \text{We can define the region }R\text{ as:} \\
& R=\left\{ \left( x,y \right):{{e}^{y}}\le x\le {{e}^{2}},\text{ }0\le y\le 2\text{ } \right\} \\
& \text{Then} \\
& \int_{1}^{{{e}^{2}}}{\int_{0}^{\ln x}{f\left( x,y \right)}dydx}=\int_{0}^{2}{\int_{{{e}^{y}}}^{{{e}^{2}}}{f\left( x,y \right)}dxdy} \\
\end{align}\]
