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