Answer
\[\frac{\sqrt{3}}{2}\]
Work Step by Step
\[\begin{align}
& \text{Let the region }R=\left\{ \left( x,y \right):0\le y\le \sec x,\text{ }0\le x\le \frac{\pi }{3}\text{ } \right\} \\
& \iint\limits_{R}{y}dA=\int_{0}^{\pi /3}{\int_{0}^{\sec x}{ydydx}} \\
& =\int_{0}^{\pi /3}{\left[ \frac{1}{2}{{y}^{2}} \right]_{0}^{\sec x}}dx \\
& =\int_{0}^{\pi /3}{\frac{1}{2}{{\sec }^{2}}x}dx \\
& =\frac{1}{2}\int_{0}^{\pi /3}{{{\sec }^{2}}x}dx \\
& =\frac{1}{2}\left[ \tan x \right]_{0}^{\pi /3} \\
& =\frac{1}{2}\tan \left( \frac{\pi }{3} \right)-\frac{1}{2}\tan \left( 0 \right) \\
& =\frac{\sqrt{3}}{2} \\
\end{align}\]
