The figure shows the parabola $y = 4 - x^2$ and the V-shaped graph of $y = |x| - 2$. The two curves enclose a region, shaded in the figure: it lies above the V and below the parabola. Part of the region is above the $x$-axis and part is below it, but area is counted as positive everywhere. Both curves are symmetric about the $y$-axis, which can halve the work. Let $A$ be the area of the region, in square units. What is the value of $3A$?

Fig. 1: The parabola y = 4 - x^2, opening downwards with its top at (0, 4), and the V-shaped graph y = |x| - 2, with its corner at (0, -2). The region above the V and below the parabola is shaded; it crosses the x-axis. Axes and a light grid are shown.