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