5. Find the number of lattice points in the plane region (excluding the boundary) bounded by the parabola $x^{2}=2 y$, the $x$-axis, and the line $x=21$.


5. When $x=2 k(k=1,2,3, \cdots, 10)$, $0<x<21$, the number of integer $y$ that satisfies $0<y<\frac{x^{2}}{2}=2 k^{2}$ is $2 k^{2}-1$; when $x=2 k-1(k=1,2,3, \cdots, 10)$, $0<x<21$, the number of integer $y$ that satisfies $0<y<\frac{x^{2}}{2}=2 k^{2}-2 k+\frac{1}{2}$ is $2 k^{2}-2 k$. Therefore, the total number of lattice points is $\sum_{k=1}^{10}\left(2 k^{2}-1\right)$ $+\sum_{k=1}^{10}\left(2 k^{2}-2 k\right)=1420$ (points).