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To solve the problem, we follow these steps:

Step 1: Setup Coordinates

• Original circle (C) (radius 1) centered at (O(0,0)).
• Points (A(1,0)) and (B(0,1)) (since (AB=\sqrt{2}), (\angle AOB=90^\circ)).

Step 2: Identify Key Intersections

• Intersection of (D_A) (center (A), radius 1) and (D_B) (center (B), radius 1): ((0,0)) and ((1,1)) (only ((0,0)) inside (C)).
• Intersection of (D_A) and (C): ((\frac{1}{2},\frac{\sqrt{3}}{2}))](#).
• Intersection of (D_B) and (C): ((\frac{\sqrt{3}}{2},\frac{1}{2})](#).

Step 3: Calculate Areas of Segments

The desired area is the sum of three segments from (D_A), (D_B), and (C):

1. Segment from (D_A) (arc (PQ), center (A)):
   Sector angle (\pi/3), segment area (\frac{\pi}{6}-\frac{\sqrt{3}}{4}).

2. Segment from (D_B) (arc (RP), center (B)):
   Same as above: (\frac{\pi}{6}-\frac{\sqrt{3}}{4}).

3. Segment from (C) (arc (QR), center (O)):
   Sector angle (\pi/6), segment area (\frac{\pi}{12}-\frac{1}{4}).

Step 4: Sum the Segments

Total area = ((\frac{\pi}{6}-\frac{\sqrt{3}}{4}) + (\frac{\pi}{6}-\frac{\sqrt{3}}{4}) + (\frac{\pi}{12}-\frac{1}{4}))

Simplify:
[=\frac{5\pi}{12} - \frac{2\sqrt{3}+1}{4}]

Answer: (\boxed{\frac{5\pi}{12} - \frac{2\sqrt{3}+1}{4}}) (or equivalent form)

But often written as (\boxed{\frac{5\pi}{12} - \frac{\sqrt{3}}{2} - \frac{1}{4}})

Final answer in boxed form: (\boxed{\frac{5\pi}{12} - \frac{2\sqrt{3} + 1}{4}})

(\boxed{\frac{5\pi}{12} - \frac{2\sqrt{3} + 1}{4}})

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