Answer
\[\frac{1}{2}-\frac{1}{2{{e}^{4}}}\]
Work Step by Step
\[\begin{align}
& \int_{0}^{2}{\int_{x}^{2}{{{e}^{-{{y}^{2}}}}}dydx} \\
& \text{Switch the order of integration using the region shown below} \\
& \int_{0}^{2}{\int_{x}^{2}{{{e}^{-{{y}^{2}}}}}dydx}=\int_{0}^{2}{\int_{0}^{y}{{{e}^{-{{y}^{2}}}}}dxdy} \\
& \text{Integrating} \\
& =\int_{0}^{2}{\left[ x{{e}^{-{{y}^{2}}}} \right]_{0}^{y}dy} \\
& =\int_{0}^{2}{y{{e}^{-{{y}^{2}}}}dy} \\
& =-\frac{1}{2}\int_{0}^{2}{{{e}^{-{{y}^{2}}}}\left( -2y \right)dy} \\
& =-\frac{1}{2}\left[ {{e}^{-{{y}^{2}}}} \right]_{0}^{2} \\
& =-\frac{1}{2}\left[ {{e}^{-{{\left( 2 \right)}^{2}}}}-{{e}^{-{{\left( 0 \right)}^{2}}}} \right] \\
& =-\frac{1}{2}\left[ {{e}^{-4}}-1 \right] \\
& =\frac{1}{2}-\frac{1}{2{{e}^{4}}} \\
\end{align}\]
