Double Integrals

Double Integrals Logo

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 (x, y) The double integral of function f (x, y) 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 \(1\)).

A function of two variables z=f(x,y) over region 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

\[a = {x_0} \lt {x_1} \lt {x_2} \lt \ldots \lt {x_i} \lt \ldots \lt {x_{m - 1}} \lt {x_m} = b.\]
Partition of a rectangular region of integration into small intervals
Figure 2.

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 \(f\left( {x,y} \right)\) 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:

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

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

\[ \begin{cases} g\left( {x,y} \right) = f\left( {x,y} \right), \;\text{if}\;\;f\left( {x,y} \right) \in R \\ g\left( {x,y} \right) = 0, \;\text{if}\;\;f\left( {x,y} \right) \notin R \end{cases} \]
Elementary general region of integration in double integral
Figure 3.

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. Region S in region R in double integral
    Figure 4.
  7. 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.
Union of regions R and S in double integral
Figure 5.

Solved Problems

Example 1.

Let \(R\) and \(S\) be non-overlapping regions (Figure \(5\)). The values of double integrals are known:

\[\iint\limits_R {f\left( {x,y} \right)dA} = 2,\;\iint\limits_R {g\left( {x,y} \right)dA} = 3,\;\iint\limits_S {f\left( {x,y} \right)dA} = 6,\;\iint\limits_S {g\left( {x,y} \right)dA} = 7.\]

Evaluate the integral \[\iint\limits_{R \cup S} {\left[ {10f\left( {x,y} \right) + 20g\left( {x,y} \right)} \right]dA} .\]


Using properties of the double integrals, we have

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