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):
-
Segment from (D_A) (arc (PQ), center (A)):
Sector angle (\pi/3), segment area (\frac{\pi}{6}-\frac{\sqrt{3}}{4}). -
Segment from (D_B) (arc (RP), center (B)):
Same as above: (\frac{\pi}{6}-\frac{\sqrt{3}}{4}). -
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}})
作者声明:本文包含人工智能生成内容。
