Example 3 Given the point set $A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leqslant\left(\frac{5}{2}\right)^{2}\right.\right\}, B=$ $\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\}$, find the number of integer points in the point set $A \cap B$.
Solution: As shown in Figure 31-3, circle $A_{1}$ and circle $B_{1}$ intersect at points $E$ and $F$. The entire figure is symmetrical about the line connecting the centers $A_{1} B_{1}$. Among them, $A \cap B$ is a crescent shape $P$ located in the lower left near the origin $O$. In $P$, the x-coordinates of the integer points can be $1, 2, 3, 4$, and the y-coordinates can be $2, 3, 4, 5$. The axis of symmetry $A_{1} B_{1}$ passes through the crescent shape $P$ and only through one integer point $C_{4}$ $(2,3)$. In the crescent shape $P$, there are 3 integer points with x-coordinate 1: $C_{1}(1,5)$, $C_{2}(1,4)$, $C_{3}(1,3)$. The symmetric points of these three points about $A_{1} B_{1}$ are $C_{5}(2,2)$, $C_{6}(3,2)$, $C_{7}(4,2)$, respectively. Therefore, there are a total of 7 integer points.