Answer
\[\int_{0}^{4}{\int_{y/2}^{\sqrt{y}}{f\left( x,y \right)}dx}dy\]
Work Step by Step
\[\begin{align}
& \int_{0}^{2}{\int_{{{x}^{2}}}^{2x}{f\left( x,y \right)}dy}dx \\
& y=2x\to x=y/2 \\
& y={{x}^{2}}\to x=\sqrt{y} \\
& \text{Using the graph to switch the order of integration} \\
& R=\left\{ \left( x,y \right):y/2\le x\le \sqrt{y},\text{ 0}\le y\le 4\text{ } \right\} \\
& \text{Then} \\
& \int_{0}^{2}{\int_{{{x}^{2}}}^{2x}{f\left( x,y \right)}dy}dx=\int_{0}^{4}{\int_{y/2}^{\sqrt{y}}{f\left( x,y \right)}dx}dy \\
\end{align}\]
