jmc

The two shaded regions combine to form a semicircle. The area of this semicircle is half the area of a circle with radius 4, or $\frac{1}{2}\cdot \pi \cdot 4^2=8\pi$. The area of the non-shaded region is equal to the area of the rectangle minus the area of the shaded region, or $8\cdot 4-8\pi =\boxed{32-8\pi }$ square units.