Answer
\[\int_{0}^{{{e}^{-2}}}{\int_{-1}^{2}{f\left( x,y \right)}dxdy}+\int_{{{e}^{-2}}}^{e}{\int_{-1}^{-\ln y}{f\left( x,y \right)}dxdy}\]
Work Step by Step
\[\begin{align}
& \int_{-1}^{2}{\int_{0}^{{{e}^{-x}}}{f\left( x,y \right)}dydx} \\
& \text{Using the graph to switch the order of integration} \\
& -1\le x\le 2,\text{ and 0}\le y\le {{e}^{-2}} \\
& -1\le x\le -\ln y,\text{ and }{{e}^{-2}}\le y\le e \\
& \text{Then} \\
& \int_{0}^{{{e}^{-2}}}{\int_{-1}^{2}{f\left( x,y \right)}dxdy}+\int_{{{e}^{-2}}}^{e}{\int_{-1}^{-\ln y}{f\left( x,y \right)}dxdy} \\
\end{align}\]
