Double Integrals Over General Regions

Double integrals are often used to measure quantities distributed across two-dimensional regions, such as rainfall over a lake, heat across a metal plate, or population density over land. In many practical situations, the region of interest does not have straight boundaries and cannot be described conveniently as a rectangle. Instead, the region may have curved or irregular edges. To evaluate integrals over such domains, the region is embedded inside a larger rectangular region where integration methods are easier to apply.

Extending Double Integrals to Irregular Regions

Let D represent an irregular region enclosed within a rectangular region R. Suppose f(x,y) describes a quantity distributed over D, such as rainfall intensity over a lake. A new function F(x,y) is defined on the entire rectangle R by

\begin{equation*}F(x,y)=\begin{cases}f(x,y), & (x,y)\in D \\0, & (x,y)\in R \text{ but not in } D\end{cases}

\end{equation*}

This construction allows integration to be performed over the rectangle R while ensuring that only values inside D contribute to the result.

Since F(x,y) = 0 outside the irregular region, the surrounding portions of the rectangle add nothing to the integral.

The double integral over the general region D is therefore written as

\begin{equation*}\iint_D f(x,y)\, dA = \iint_R F(x,y)\, dA\end{equation*}

Type I and Type II Regions

Irregular regions are commonly classified into two forms. A type I region is bounded vertically and can be expressed as

\begin{equation*}D=\{(x,y)\mid a\le x\le b,\; g_1(x)\le y\le g_2(x)\}\end{equation*}

A type II region is bounded horizontally and is written as

\begin{equation*}D=\{(x,y)\mid c\le y\le d,\; h_1(y)\le x\le h_2(y)\}\end{equation*}

These descriptions convert integration over curved regions into iterated integrals with variable limits, enabling systematic evaluation of quantities distributed over irregular domains.