# Definition and Properties of Double Integrals

### Definition of Double Integral

The definite integral can be extended to functions of more than one variable. Consider, for example, a function of two variables $$z = f\left( {x,y} \right).$$ The double integral of function $$f\left( {x,y} \right)$$ is denoted by

$\iint\limits_R {f\left( {x,y} \right)dA},$

where $$R$$ is the region of integration in the $$xy$$-plane.

If the definite integral $$\int\limits_a^b {f\left( x \right)dx}$$ of a function of one variable $${f\left( x \right)} \ge 0$$ is the area under the curve $${f\left( x \right)}$$ from $$x = a$$ to $$x = b,$$ then the double integral is equal to the volume under the surface $$z = f\left( {x,y} \right)$$ and above the $$xy$$-plane in the region of integration $$R$$ (Figure $$1$$).

As in the case of integral of a function of one variable, a double integral is defined as a limit of a Riemann sum.

If the region $$R$$ is a rectangle $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ (Figure $$2$$), we can subdivide $$\left[ {a,b} \right]$$ into small intervals with a set of numbers $$\left\{ {{x_0},{x_1}, \ldots ,{x_m}} \right\}$$ so that

${a }={ {x_0} \lt {x_1} \lt {x_2} \lt \ldots} \lt{ {x_i} \lt \ldots } \lt { {x_{m – 1}} \lt {x_m} }={ b.}$

Similarly, a set of numbers $$\left\{ {{y_0},{y_1}, \ldots ,{y_n}} \right\}$$ is said to be a partition of $$\left[ {c,d} \right]$$ along the $$y$$-axis, if

${c }={ {y_0} \lt {y_1} \lt {y_2} \lt \ldots} \lt{ {y_j} \lt \ldots } \lt { {y_{n – 1}} \lt {y_n} }={ d.}$

The Riemann sum of a function f (x,y) over this partition of $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ is

$\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {f\left( {{u_i},{v_j}} \right)\Delta {x_i}\Delta {y_j}} } ,$

where $${\left( {{u_i},{v_j}} \right)}$$ is some point in the rectangle $$\left( {{x_{i – 1}},{x_i}} \right)$$ $$\times \left( {{y_{j – 1}},{y_j}} \right)$$ and $$\Delta {x_i} = {x_i} – {x_{i – 1}},$$ $$\Delta {y_j} = {y_j} – {y_{j – 1}}.$$

We then define the double integral of a function $${f\left( {x,y} \right)}$$ in the rectangular region $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ to be the limit of the Riemann sum as maximum values of $$\Delta {x_i}$$ and $$\Delta {y_j}$$ approach zero:

$\require{AMSmath.js} {\iint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right]} {f\left( {x,y} \right)dA} } = {\lim\limits_{\substack{ \text{max}\,\Delta {x_i} \to 0\\ \text{max}\,\Delta {y_j} \to 0}} \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {f\left( {{u_i},{v_j}} \right)}}}\kern0pt{{{\Delta {x_i}\Delta {y_j}} } .}$

To define the double integral over a bounded region $$R$$ other than a rectangle, we choose a rectangle $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ that contains $$R$$ (Figure $$3\text{),}$$ and define the function $${g\left( {x,y} \right)}$$ so that

$\begin{cases} {g\left( {x,y} \right) = f\left( {x,y} \right), \;}\kern-0.3pt {\text{if}\;\;f\left( {x,y} \right) \in R }\\ {g\left( {x,y} \right) = 0, \;}\kern-0.3pt {\text{if}\;\;f\left( {x,y} \right) \notin R} \end{cases}$

Then the double integral of the function $${f\left( {x,y} \right)}$$ over a general region $$R$$ is defined to be

${\iint\limits_R {f\left( {x,y} \right)dA} } = {\iint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right]} {g\left( {x,y} \right)dA}.}$

### Properties of Double Integrals

The double integral satisfies the following properties:

1. $${\iint\limits_R {\left[ {f\left( {x,y} \right) + g\left( {x,y} \right)} \right]dA} }$$ $$= {\iint\limits_R {f\left( {x,y} \right)dA} }$$ $$+{ \iint\limits_R {g\left( {x,y} \right)dA} ;}$$
2. $${\iint\limits_R {\left[ {f\left( {x,y} \right) – g\left( {x,y} \right)} \right]dA} }$$ $$= {\iint\limits_R {f\left( {x,y} \right)dA} }$$ $$-{ \iint\limits_R {g\left( {x,y} \right)dA} ;}$$
3. $$\iint\limits_R {kf\left( {x,y} \right)dA}$$ $$= k\iint\limits_R {f\left( {x,y} \right)dA},$$ where $$k$$ is a constant;
4. If $${f\left( {x,y} \right)} \le {g\left( {x,y} \right)}$$ on $$R,$$ then $$\iint\limits_R {f\left( {x,y} \right)dA}$$ $$\le \iint\limits_R {g\left( {x,y} \right)dA} ;$$
5. If $${f\left( {x,y} \right)} \ge 0$$ on $$R$$ and $$S \subset R$$ (Figure $$4$$), then $$\iint\limits_S {f\left( {x,y} \right)dA}$$ $$\le \iint\limits_R {f\left( {x,y} \right)dA} ;$$
6. If $${f\left( {x,y} \right)} \ge 0$$ on $$R$$ and $$R$$ and $$S$$ are non-overlapping regions (Figure $$5$$), then
${\iint\limits_{R \cup S} {f\left( {x,y} \right)dA} }={\iint\limits_R {f\left( {x,y} \right)dA} }+{ \iint\limits_S {f\left( {x,y} \right)dA}.}$
Here $${R \cup S}$$ is the union of these two regions.
Figure 4.
Figure 5.

## Solved Problems

### Example 1.

Let $$R$$ and $$S$$ be non-overlapping regions (Figure $$5$$). The values of double integrals are known:
${\iint\limits_R {f\left( {x,y} \right)dA} = 2,\;\;}\kern-0.3pt {\iint\limits_R {g\left( {x,y} \right)dA} = 3,\;\;}\kern-0.3pt {\iint\limits_S {f\left( {x,y} \right)dA} = 6,\;\;}\kern-0.3pt {\iint\limits_S {g\left( {x,y} \right)dA} = 7.}$
Evaluate the integral $$\iint\limits_{R \cup S} {\left[ {10f\left( {x,y} \right) }\right.}$$ $$+\,{\left.{ 20g\left( {x,y} \right)} \right]dA} .$$

Solution.

Using properties of the double integrals, we have

${\iint\limits_{R \cup S} {\left[ {10f\left( {x,y} \right) }\right.}+{\left.{ 20g\left( {x,y} \right)} \right]dA} } = {\iint\limits_{R \cup S} {10f\left( {x,y} \right)dA} }+{ \iint\limits_{R \cup S} {20g\left( {x,y} \right)dA} } = {{10\iint\limits_{R \cup S} {f\left( {x,y} \right)dA} }+{ 20\iint\limits_{R \cup S} {g\left( {x,y} \right)dA} }} = {{10\left[ {\iint\limits_R {f\left( {x,y} \right)dA} }\right.}+{\left.{ \iint\limits_S {f\left( {x,y} \right)dA} } \right] }} + {{20\left[ {\iint\limits_R {g\left( {x,y} \right)dA} }\right.}+{\left.{ \iint\limits_S {g\left( {x,y} \right)dA} } \right] }} = {10\left( {2 + 6} \right) }+{ 20\left( {3 + 7} \right) }={ 280.}$