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