Definition and Properties of Double Integrals


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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 \(1\)).

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

\[a = {x_0} \lt {x_1} \lt {x_2} \lt \ldots \lt {x_i} \lt \ldots \lt {x_{m - 1}} \lt {x_m} = b.\]
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

\[ \begin{cases} g\left( {x,y} \right) = f\left( {x,y} \right), \;\text{if}\;\;f\left( {x,y} \right) \in R \\ g\left( {x,y} \right) = 0, \;\text{if}\;\;f\left( {x,y} \right) \notin R \end{cases} \]
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. 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.
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

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