The shaded region is the difference of two $120^\circ$ sectors (a $120^\circ$ sector is $\frac{1}{3}$ of a circle).

Outer radius $XZ = XY + YZ = 6 + 16 = 22$; inner radius $XY = 6$. $A = \frac{120}{360}\pi\left(22^2 - 6^2\right) = \frac{1}{3}\pi(484 - 36)$ $= \frac{448\pi}{3} \approx 469$