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To solve the problem of finding the area of the intersection of two circles (assuming this is the shaded region, given common problem contexts), here's the step-by-step solution:

Key Observations

Assume two circles of radius (r=5) with centers separated by (d=8). The intersection area is symmetric, so we calculate the segment area of one circle and double it.

Step 1: Calculate Chord Length & Height

• Midpoint (M) of the common chord (AB) divides the center line into (OM=4) (since (d=8)).
• Chord length (AB = 2\sqrt{r^2 - OM^2} = 2\sqrt{25-16}=6).

Step 2: Sector Area

• Angle (\theta = \angle AOB = 2\arccos\left(\frac{OM}{r}\right)=2\arccos\left(\frac{4}{5}\right)).
• Sector area (OAB = \frac{1}{2}r^2\theta = \frac{1}{2}(25)(2\arccos\frac{4}{5})=25\arccos\frac{4}{5}).

Step 3: Triangle Area

• Area of (\triangle OAB = \frac{1}{2} \times AB \times OM = \frac{1}{2}\times6\times4=12).

Step 4: Intersection Area

• Segment area = Sector area - Triangle area = (25\arccos\frac{4}{5}-12).
• Total intersection area = (2\times) segment area = (50\arccos\frac{4}{5}-24).

Exact Answer: (\boxed{50\arccos\left(\frac{4}{5}\right)-24})

Numerical Approximation: (\approx8.18) (if rounded to two decimals, but exact form is preferred).

(\boxed{50\arccos\left(\frac{4}{5}\right)-24})

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