Answer
\[\int_{1}^{4}{\int_{1}^{\frac{2}{\sqrt{y}}}{f\left( x,y \right)}dxdy}\]
Work Step by Step
\[\begin{align}
& \int_{1}^{2}{\int_{1}^{4/{{x}^{2}}}{f\left( x,y \right)}dydx} \\
& y=\frac{4}{{{x}^{2}}}\to x=\frac{2}{\sqrt{y}} \\
& \text{Using the graph to switch the order of integration} \\
& \text{We can define the region }R\text{ as:} \\
& R=\left\{ \left( x,y \right):1\le x\le \frac{2}{\sqrt{y}},\text{ }1\le y\le 4\text{ } \right\} \\
& \text{Then} \\
& \int_{1}^{2}{\int_{1}^{4/{{x}^{2}}}{f\left( x,y \right)}dydx}=\int_{1}^{4}{\int_{1}^{\frac{2}{\sqrt{y}}}{f\left( x,y \right)}dxdy} \\
\end{align}\]
