In a textbook I am consulting re: Green's Theorem, they illustrate with the following example:
Evaluate $\int_C y^2\, \text{d}x + 3 xy \, \text{d}y$, where $C$ is the boundary of the semiannular region $D$ in the upper half-plane between the circles $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$.
The first line of the solution says
Notice that although $D$ is not simple, the $y$-axis divides it into two simple regions.
But I don't see how the semiannular region isn't simple: I don't see any closed curve within $D$ that could not be continuously shrunk to a point without passing outside of $D$. What am I missing?
The remark doesn't seem to play any role in the rest of the solution: they solve exactly as I would, but I'd still like to understand this claim about the domain.
I believe the author is talking about Type I and Type II (also known as $x$-simple and $y$-simple) regions used to define regions for integration with Green's theorem, i.e. regions of the form $\{(x,y) \mid a \leq x \leq b, f(x) \leq y \leq g(x)\}$ and $\{(x,y) \mid f(y) \leq x \leq g(y), a \leq y \leq b\}$, and not about it being simply connected.
If this is the case, what was meant by this sentence was: Although the region is neither $x$-simple nor $y$-simple ...