Circles centered at A and B each have radius 2, as shown. Point O is the midpoint of \overline{A B}, and O A=2 \sqrt{2}. Segments O C and O D are tangent to the circles centered at A and B, respectively, and \overline{E F} is a common tangent. What is the area of the shaded region E C O D F ?

**Answer Choices**

A. \dfrac{8 \sqrt{2}}{3}

B. 8 \sqrt{2}-4-\pi

C. 4 \sqrt{2}

D. 4 \sqrt{2}+\dfrac{\pi}{8}

E. 8 \sqrt{2}-2-\dfrac{\pi}{2}