Let the area of the region \ ( x, y ) : 0 ≤ x ≤ 3, 0 ≤ y ≤ min \ x...

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,

$A = {∫}_{0}^{2}{x}^{2} + 2 d x + {∫}_{2}^{3}2x + 2 d x$

$⇒ A = {\left[\dfrac{{x}^{3}}{3} + 2x\right]}_{0}^{2} + {\left[{x}^{2} + 2x\right]}_{2}^{3}$

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

$⇒ A = \dfrac{20}{3} + 7 = \dfrac{41}{3}$

$⇒ 12A = 164$