Answer
\[\ln \sqrt[4]{2}\]
Work Step by Step
\[\begin{align}
& \text{Let the region }R=\left\{ \left( x,y \right):0\le y\le {{x}^{2}},\text{ }0\le x\le \sqrt{\pi }\text{/2 } \right\} \\
& \iint\limits_{R}{\left( x+y \right)}dA=\int_{0}^{\sqrt{\pi }/2}{\int_{0}^{{{x}^{2}}}{x{{\sec }^{2}}ydydx}} \\
& \text{Integrating} \\
& =\int_{0}^{\sqrt{\pi }/2}{\left[ x\tan y \right]_{0}^{{{x}^{2}}}}dx \\
& =\int_{0}^{\sqrt{\pi }/2}{x\tan {{x}^{2}}}dx \\
& =\frac{1}{2}\int_{0}^{\sqrt{\pi }/2}{2x\tan {{x}^{2}}}dx \\
& =\frac{1}{2}\left[ \ln \left| \sec {{x}^{2}} \right| \right]_{0}^{\sqrt{\pi }/2} \\
& =\frac{1}{2}\left[ \ln \left| \sec {{\left( \frac{\sqrt{\pi }}{2} \right)}^{2}} \right|-\ln \left| \sec {{\left( 0 \right)}^{2}} \right| \right] \\
& =\frac{1}{2}\left[ \ln \left| \sec \left( \frac{\pi }{4} \right) \right|-\ln \left| 1 \right| \right] \\
& =\frac{1}{2}\left[ \ln \left| \sqrt{2} \right| \right] \\
& =\ln \sqrt[4]{2} \\
\end{align}\]
