Answer
\[\int_{-3}^{4}{\int_{2{{x}^{2}}}^{2x+24}f(x,y){dy}dx}\]
Work Step by Step
\[\begin{align}
& y=2x+24,\text{ }y=2{{x}^{2}} \\
& \text{Let }y=y \\
& 2x+24=2{{x}^{2}} \\
& 2{{x}^{2}}-2x-24=0 \\
& {{x}^{2}}-x-12=0 \\
& x=-3,\text{ }x=4 \\
& \text{From the graph, the region }R\text{ is} \\
& R=\left\{ \left( x,y \right):2{{x}^{2}}\le y\le 2x+24,-3\le x\le 4 \right\} \\
& \text{Then,} \\
& \int_{-3}^{4}{\int_{2{{x}^{2}}}^{2x+24}f(x,y){dy}dx} \\
\end{align}\]
