Corollary 2

To really be sure that the double integral corresponds to the volume, we need to know that it is the only operation that has the Addition and Cylinder Properties. To make this precise, we introduce the notion of a volume function.

We suppose f(x, y) is continuous at every point of an open region D₀, and consider subregions D of D₀. A volume function for f is a function B which assigns a real number B(D) to each subregion D of D₀ and has the Addition Property

B(D) = B(D₁) + B(D₂)

and the Cylinder Property

mA ≤ B(D) ≤ MA,

where m is the minimum and M the maximum value of f on D.