6. Answer: (A).
The area enclosed by $A D, D E$ and $\widehat{A E}$ is $\frac{\pi\left(2^{2}\right)}{8}-1=\frac{\pi}{2}-1$.
The area of the wedge $E D F$ is $\frac{\pi(2-\sqrt{2})^{2}}{4}=\left(\frac{3}{2}-\sqrt{2}\right) \pi$.
So the area of the egg is: $\frac{\pi}{2}+1+2 \times\left(\frac{\pi}{2}-1\right)+\left(\frac{3}{2}-\sqrt{2}\right) \pi=(3-\sqrt{2}) \pi-1$.
6. Consider the following egg shaped curve. $A B C D$ is a circle of radius 1 centred at $O$. The arc $\overparen{A E}$ is centred at $C, \overparen{C F}$ is centred at $A$ and $\overparen{E F}$ is centred at $D$.

What is the area of the region enclosed by the egg shaped curve?
(A) $(3-\sqrt{2}) \pi-1$;
(B) $(3-\sqrt{2}) \pi$;
(C) $(3+\sqrt{2}) \pi+1$;
(D) $(3-2 \sqrt{2}) \pi ;$
(E) $(3-2 \sqrt{2}) \pi+1$.