Answer
\[\frac{1}{2}\left( e-1 \right)\]
Work Step by Step
\[\begin{align}
& \int_{0}^{1}{\int_{y}^{1}{{{e}^{{{x}^{2}}}}}dxdy} \\
& \text{Using the graph to switch the order of integration} \\
& R=\left\{ \left( x,y \right):0\le y\le x,\text{ 0}\le x\le 1\text{ } \right\} \\
& \text{Then}, \\
& \int_{0}^{1}{\int_{y}^{1}{{{e}^{{{x}^{2}}}}}dxdy}=\int_{0}^{1}{\int_{0}^{x}{{{e}^{{{x}^{2}}}}}dydx} \\
& \text{Integrating} \\
& =\int_{0}^{1}{\left[ {{e}^{{{x}^{2}}}}y \right]_{0}^{x}dx} \\
& =\int_{0}^{1}{x{{e}^{{{x}^{2}}}}dx} \\
& =\frac{1}{2}\left[ {{e}^{{{x}^{2}}}} \right]_{0}^{1} \\
& =\frac{1}{2}\left[ {{e}^{1}}-{{e}^{0}} \right] \\
& =\frac{1}{2}\left( e-1 \right) \\
\end{align}\]
