Let the area of the region $\{(x,y): 0 \leq x \leq 3, 0 \leq y \leq \min\{x^2+2, 2x+2\}\}$ be $A$. Then…

Given,

$0\le y\le \min \left\{x^2+2, 2x+2\right\}, x\in \left[0,3\right]$

Now, plotting the diagram of the above expression we get,

Now, from above diagram we get,

$A={\int }_0^2{x}^2+2\mathit{d}x+{\int }_2^{3}2x+2\mathit{d}x$

$\Rightarrow A={\left[\frac{x^3}{3}+2x\right]}_0^2+{\left[x^2+2x\right]}_2^3$

$\Rightarrow A=\left[\frac{8}{3}+4\right]+\left[9+6-4-4\right]$

$\Rightarrow A=\frac{20}{3}+7=\frac{41}{3}$

$\Rightarrow 12A=164$