jmc

The number $\pi$ is between $3.14$ and $3.15$, so $-8\pi$ is between $-8(3.15)=25.2$ and $-8(3.14)=25.12$. Likewise, $10\pi$ is between $31.4$ and $31.5$. This suffices to establish that the integers $n$ between $-8\pi$ and $10\pi$ are precisely $$-25,-24,-23,-22,\dots ,28,29,30,31.$$ There are $25$ negative integers in this list, $31$ positive integers, and one more integer ($0$), making $\boxed{57}$ integers in total.