1. The Gate Widened: From Rectangles to Curve-Bounded Regions
Recall that a domain in several-variable calculus is itself a region in the plane. The skill you built describing domains (e.g., "all $(x,y)$ with $x^2+y^2 \le 4$") is exactly the skill you need here. Every double integral over a general region begins with reading the region.
The Two Canonical Region Types
A region is vertically simple (Type I) if every vertical line that meets the region enters through a bottom curve and exits through a top curve:
$$R = \{(x,y) \mid a \le x \le b,\; g_1(x) \le y \le g_2(x)\}$$Think: you sweep $x$ from left to right (constants $a$ to $b$), and for each $x$ the variable $y$ runs from the lower boundary $g_1(x)$ up to the upper boundary $g_2(x)$.
A region is horizontally simple (Type II) if every horizontal line that meets the region enters from the left curve and exits through the right curve:
$$R = \{(x,y) \mid c \le y \le d,\; h_1(y) \le x \le h_2(y)\}$$Think: you sweep $y$ from bottom to top (constants $c$ to $d$), and for each $y$ the variable $x$ runs from the left boundary $h_1(y)$ to the right boundary $h_2(y)$.
Interactive Visualization: Type I vs Type II Region
Type I (Vertically Simple)
Type II (Horizontally Simple)