Example 3 As shown in Figure $2, \odot O$ has a radius of $2, \odot O$ contains a point $P$ that is 1 unit away from the center $O$, and a chord $AB$ passing through point $P$ forms a segment with the minor arc $\overparen{AB}$. Find the minimum value of the area of this segment. (13th Junior High School Mathematics Competition in Jiangsu Province)