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Let the equation of the large circle be $\"\"$ .

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Then the equation of the small circle is $\"\"$ .

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where $\"\"$ is the distance between the centers of the circles.

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Area of the crescent-shaped region is $\"\"$ .

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$\"\"$

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$\"\"$ .

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Evaluate the integral $\"\"$ .

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$\"\"$

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$\"\"$ .

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Evaluate the integral $\"\"$ .

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Apply the special integral formula

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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Evaluate the integral $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Area of the crescent-shaped region is,

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$ .