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Home Multiple Integrals Double Integrals Properties of a Volume Function
 
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Properties of a Volume Function

Instead of closed intervals [u, v] in the line, we deal with closed regions D in the plane. A volume function for f(x, y) is a function B, which assigns a real number B(D) to each closed region D, and has the following two properties: Addition Property and Cylinder Property.

ADDITION PROPERTY

If D is divided into two regions D1 and D2 which meet only on a common boundary curve, then

B(D) = B(D1) + B(D2).

(Intuitively, the volume over D is the sum of the volumes over D1 and D2.) This property is illustrated in Figure 12.1.2(a).

12_multiple_integrals-6.gif

Figure 12.1.2(a) Addition Property

 

CYLINDER PROPERTY

Let in and M be the minimum and maximum values of f(x, y) on D and let A be the area of D. Then

mA ≤ B(D) ≤ MA.

(Intuitively, the volume over D is between the volumes of the cylinders over D of height m and M. This corresponds to the Rectangle Property for single integrals.) This property is illustrated in Figure 12.1.2(b).

12_multiple_integrals-7.gif

Figure 12.1.2 (b) Cylinder Property

We shall see at the end of this section that the double integral

12_multiple_integrals-5.gif

is the unique volume function for a continuous function f(x, y). The double integral will be constructed using double Riemann sums, just as the single integral was constructed from single Riemann sums.
Home Multiple Integrals Double Integrals Properties of a Volume Function

Last Update: 2006-11-05