Type I, Type II, and elementary regions for Green's theorem

A compact Type I region is

$D=\{(x,y):a\le x\le b,\alpha (x)\le y\le \beta (x)\},$

where $a<b$, the continuous piecewise-$C^1$ functions $\alpha ,\beta$ satisfy $\alpha \le \beta$, and $\alpha <\beta$ on $(a,b)$. A compact Type II region is defined analogously by continuous piecewise-$C^1$ functions $\lambda \le \rho$ on $[c,d]$:

$D=\{(x,y):c\le y\le d,\lambda (y)\le x\le \rho (y)\}\mathrm{.}$

An elementary Green region admits both descriptions. By A region between two continuous graphs is Jordan measurable, and a continuous integrand extending to its closure integrates by vertical sections, each Type I description is compact and Jordan measurable.

A finite elementary Green region is a nonempty finite union $D=D_1\cup \cdots \cup D_N$ of elementary Green regions with pairwise disjoint interiors. Pairwise intersections must be finite unions of complete shared boundary arcs and endpoints, and every positive-length internal arc must belong to exactly two pieces with opposite induced orientations. This supplied decomposition is part of the data; it is not inferred from an arbitrary closed curve.