By symmetry, we only need to consider the part of the area above the $x$-axis. Suppose line $y = k$ intercepts the right branch of the hyperbola and line $x = 100$ at points $A_k$ and $B_k$ ($k = 1, 2, \dots, 99$), respectively. Then the number of integral points within the segment $A_k B_k$ is $99 - k$. Therefore, the number of integral points within the area above the $x$-axis is
$$\sum_{k=1}^{99} (99 - k) = 99 \times 49 = 4851.$$
Finally, we obtain the total number of integral points within the whole area as $2 \times 4851 + 98 = 9800$. $\square$