Example 4 Find the number of integer points that satisfy the system of inequalities: $\left\{\begin{array}{l}y \leqslant 3 x \\ y \geqslant \frac{1}{3} x \\ x+y \leqslant 100\end{array}\right.$
Solution: The number of integer points in the region (including the boundary) enclosed by the two coordinate axes and the line $x+y=100$ is: $1+2+3+\cdots+101=5151$.

The number of integer points in the region (excluding the boundary $y=\frac{1}{3} x$) enclosed by $y=\frac{1}{3} x$, $x+y=100$, and the $x$-axis is: $(1+1+1+1)+(2+2+2+2)+\cdots+(25+25+25+25)=1300$.

The number of integer points in the region (excluding the boundary $y=3 x$) enclosed by $y=3 x$, $x+y=100$, and the $y$-axis is also 1300.
Therefore, the number of integer points that satisfy the given conditions is $5151-2 \times 1300=2551$.