Consider the rectangle with vertices at $(5,4),$ $(5,-4),$ $(-5,4),$ $(-5,-4)$. How many integer coordinates will be strictly inside the rectangular region?
I need to count how many pairs of integers $(x,y)$ satisfy $-5<x<5$ and $-4<y<4$. For each fixed value of $x$ between $-5$ and $5$, there are $7$ possible values of $y$ that are strictly between $-4$ and $4$. For example, if $x=-3$, then $y$ can be $-3,-2,-1,0,1,2,3$. There are $9$ possible values of $x$ that are strictly between $-5$ and $5$, namely $-4,-3,-2,-1,0,1,2,3,4$. Therefore, the total number of integer coordinates inside the rectangle is $9\times 7=63$. 