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