To find the area of the region enclosed between the circles given by the equations \(x^2 + y^2 = 4\) and \(x^2 + (y - 2)^2 = 4\), we proceed step by step:
Identify the Centers and Radii of the Circles:
The first circle \(x^2 + y^2 = 4\) is centered at the origin \((0, 0)\) with a radius of \(2\).
The second circle \(x^2 + (y - 2)^2 = 4\) is centered at \((0, 2)\) with a radius of \(2\).
Determine the Distance Between the Centers:
Using the distance formula, the distance between the centers \((0, 0)\) and \((0, 2)\) is: \(\sqrt{(0-0)^2 + (2-0)^2} = 2\).
Check Intersection:
Since the distance between centers is equal to the radii of the circles (both 2), the circles touch each other externally.
Calculate the Area of Intersection:
The points where the circles intersect form a lens-shaped area.
Formula for the area of two intersecting circles for a lens region can be derived through geometry or calculus. For this particular setup, you can visualize it involving sectors and triangles formed by the radii and the line connecting the intersection points.
The area \((A)\) of this intersection or the lens-shaped region can be computed as \(\frac{2}{3}(4\pi - 3\sqrt{3})\).
Conclusion:
Therefore, the area of the region enclosed between the two circles is \(\frac{2}{3}(4\pi - 3\sqrt{3})\).
Hence, the correct answer is \(\frac{2}{3}(4\pi - 3\sqrt{3})\).
