Let $D$ be the region delimited by
$$\partial D:
\begin{cases}
C_1: x^2 + y^2 = 5^2\\
C_2:(x-2)^2+y^2= 1\\
C_3:(x+2)^2+y^2 = 1\\
C_4: x^2+(y-2)^2= 1\\
C_5: x^2+(y+2)^2= 1
\end{cases}
$$
I've sketched this and the region looks like this (the inside of the big circle, intersection the outside of the small circles should be shaded):
So the boundary of the region is not a closed curve... but I'm asked to verify Green's theorem for this region anyway. I don't understand how the theorem applies, could someone explain this?
E: From the comments it looks like I should do $$\iint_D (Q_x-P_y)dA=\oint_{C_1} Fds-\sum_{i\ge 2} \oint_{C_i}Fds$$
Is this correct? I don't see how this follows from Green's theorem's statement (using wikipedia for reference).
This page has a good explanation of the technique and way to think about the multiple boundaries.
This page goes into more detail about why the technique works.