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Addition Property

ADDITION PROPERTY

Let D be divided into two regions D1 and D2 which meet only on a common boundary as in Figure 12.1.13.

Then

12_multiple_integrals-36.gif

12_multiple_integrals-37.gif 12_multiple_integrals-38.gif

Figure 12.1.13 (a) and (b)

Interpreting the double integral as a volume, the Addition Property says that the volume of the solid over D is equal to the sum of the volume over D1 and the volume over D2, as shown in Figure 12.1.14.

A continuous function z = f(x,y) always has a minimum and maximum value on a closed region D. The proof is similar to the one-variable case.

                                                                 

12_multiple_integrals-41.gif12_multiple_integrals-42.gif

Figure 12.1.14(a) and (b):Addition Property
Home Multiple Integrals Double Integrals Infinite Double Riemann Sum Addition Property

Last Update: 2006-11-05