A region $D\subset {\mathbb{R}}^2$ where every horizontal or vertical line intersects $D$ in a line segment

Simple Region

A region $R\subset {\mathbb{R}}^2$ is simple if it is x-simple and y-simple.

$$R=\{(x,y):{\varphi }_1(x)\le y\le {\varphi }_2(x)\text{ and }{\psi }_1(y)\le x\le {\psi }_2(y)\}$$

Intuition

There is a pair of curves bounding x and a pair of curves bounding y