smc

We want to find the number of lattice points in and on the boundary of the shaded region in the diagram. To do this, we will look at the integer values of $x$ from $0$ to $4$. At a given value of $x$, the amount of lattice points in the region is $x^2+1$, because all of the integers from $0$ up to and including $x^2$ are in the region. Thus, evaluating this expression at at $x=0,1,2,3,$ and $4$ and adding the results together, we see that the number of lattice points is $1+2+5+10+17=35$, so our answer is $\boxed{35}$.