$37 \cdot 11$ A point with integer (including zero) coordinates is called a lattice point (or integer point). The region enclosed by the $x$-axis, the line $x=4$, and the parabola $y=x^{2}$ (including the boundaries) contains a total number of lattice points:
(A) 24.
(B) 35.
(C) 34.
(D) 30.
(E) Cannot be determined.
(8th American High School Mathematics Examination, 1957)
[Solution] The x-coordinates of the lattice points in the above region can only be $0,1,2,3,4$. For $x=k$, the number of lattice points is $k^{2}+1$ (note $y=x^{2}$).
Therefore, the total number of lattice points is $\sum_{k=0}^{4}\left(k^{2}+1\right)=35$. Hence, the answer is $(B)$.