Answer
$2\pi-4~units^2$
Work Step by Step
The shaded region is a circle. To find the radius, we use Thales's theorem, which states that if we have distinct points A, B, C on a circle and AC is the diameter, then we have a 90 degree (right) angle at B. In other words, the shape inside the circle is a square with right angles. Hence, by applying the Pythagorean theorem with the sides $a=b=2$, we can find the diameter (hypotenuse):
$c^2=a^2+b^2=2^2+2^2=8$
$c=\sqrt{8}=2\sqrt{2}$
Thus, the radius is $c/2=\sqrt{2}$. Now, we can calculate the area of the circle as:
$$ A=(\sqrt{ 2})^2\pi=2 \pi$$
The final area is the circle area minus the square area:
$shaded~area=2\pi-2^2=2\pi-4~units^2$
