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Uniqueness Theorem
UNIQUENESS THEOREM The double integral ∫∫D f(x, y) dA is the only volume function for f. That is, if B is a function which has the Addition and Cylinder Properties, then
Given a continuous function f such that f(x, y) ≥ 0 for all (x, y), the function V(D) = volume over D certainly has the Addition and Cylinder Properties. Thus we are justified in defining the volume as the double integral. DEFINITION Let f(x, y) ≥ 0 for (x, y) in D. Then the volume over D between z = 0 and z = f(x, y) is the double integral
When f(x, y) is the constant 1, we have
That is, the area of D is equal to the volume of the cylinder with base D and height 1, as in Figure 12.1.17. Given any unit of length (say meters), if the height is one meter then the area is in square meters and the volume has the same value but in cubic meters.
Figure 12.1.17
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for every D.
Last Update: 2006-11-05