The area of the region enclosed between the circles $$x^{2}+y^{2}=4 \text{ and } x^{2}+(y-2)^{2}=4$$ is

$$x^2+y^2=4\quad(C_1),\qquad x^2+(y-2)^2=4\quad(C_2)$$

$$Centers:((0,0)),((0,2)),radius(r=2).$$
Distance between $$centers(d=2).$$

Area of intersection of two equal circles:
$$A=2r^2\cos^{-1}!\left(\frac{d}{2r}\right)-\frac{d}{2}\sqrt{4r^2-d^2}$$

Substitute $$(r=2,d=2):$$
$$A=2(4)\cos^{-1}\left(\frac{1}{2}\right)-1\cdot\sqrt{16-4}$$

$$=8\cdot\frac{\pi}{3}-\sqrt{12}$$
$$=\frac{8\pi}{3}-2\sqrt{3}$$

Factor:
$$=\frac{2}{3}(4\pi-3\sqrt{3})$$