Geometry Problem from Singapore ( 2 )

Split up the region and rotate one piece $180°$ as follows:

The area of the shaded region is now:

$= A_{\text{big quartercircle}} - A_{\text{small quartercircle}} - 3A_{\text{square}} - 2A_{\text{triangle}}$

$= \frac{1}{4}\pi 12^2 - \frac{1}{4}\pi 4^2 - 3 \cdot 4 \cdot 4 - 2 \cdot \frac{1}{2} \cdot 4 \cdot 4$

$= 32 \pi - 64$

$= 2^5 \pi - 2^6$ .

Therefore, $a = 5$ , $b = 6$ , and $a + b = \boxed{11}$ .