A plane contains points A and B with A B=1. Let S be the union of all disks of radius 1 in the plane that cover \overline{A B}. What is the area of S ?
Answer Choices
A. 2\pi+\sqrt{3}
B. \dfrac{8\pi}{3}
C. 3 \pi - \dfrac{\sqrt{3}}{2}
D. \dfrac{10\pi}{3} - \sqrt{3}
E. 4 \pi - 2 \sqrt{3}