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

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

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

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

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:

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

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

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.

Solved Problems

Solution.

Using properties of the double integrals, we have