Problem #2184


A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$?

[asy] unitsize(1.5cm); defaultpen(.8);  draw( circle( (0,0), 2 ) ); draw( (-2,0) -- (2,0) ); draw( (0,-2) -- (0,2) );  pair D = intersectionpoint( circle( (0,0), 2 ), (1,0) -- (1,2) ); pair Ep = intersectionpoint( circle( (0,0), 2 ), (0,1) -- (2,1) ); draw( (1,0) -- D ); draw( (0,1) -- Ep );  filldraw( (1,1) -- arc( (0,0),Ep,D ) -- cycle, mediumgray, black );  label("$O$",(0,0),SW); label("$A$",(1,0),S); label("$C$",(0,1),W); label("$B$",(1,1),SW); label("$D$",D,N); label("$E$",Ep,E); [/asy]

$\mathrm{(A) \ } \frac{\pi}{3}+1-\sqrt{3}\qquad \mathrm{(B) \ } \frac{\pi}{2}(2-\sqrt{3})\qquad \mathrm{(C) \ } \pi(2-\sqrt{3})\qquad \mathrm{(D) \ } \frac{\pi}{6}+\frac{\sqrt{3}+1}{2}\qquad \mathrm{(E) \ } \frac{\pi}{3}-1+\sqrt{3}$

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