Mathematics

Problem

Consider the function

f, g: \mathbb{R} \to \mathbb{R}

defined by

f(x) = x^2 + \frac{5}{12}

and

g(x) = \begin{cases} 2\left(1 - \frac{4|x|}{3}\right), & |x| \le \frac{3}{4}, \\ 0, & |x| > \frac{3}{4} \end{cases}

. If

\alpha

is the area of the region

\left\{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| \le \frac{3}{4}, 0 \le y \le \min\{f(x), g(x)\}\right\}

, then the value of

9\alpha

is \_\_\_\_\_\_.