vertérfogat

Vertérfogat is a term found in some Nordic-language mathematical literature that denotes the volume of a three-dimensional object when it is analyzed via vertical cross-sections. The concept emphasizes summing or integrating the areas of slices taken along a vertical axis to obtain the total volume.

Formally, let S be a solid whose vertical coordinate is z. For each z in the interval [z0, z1], the cross-section A(z) is the set of points (x, y) for which (x, y, z) belongs to S, and A(z) is its area. The vertical volume is then given by V = ∫_{z0}^{z1} A(z) dz. If the solid can be described as all points (x, y, z) with (x, y) in a base region D and 0 ≤ z ≤ h(x, y), then V = ∬_D h(x, y) dx dy. This formulation follows from applying Fubini’s theorem to integrate over height first, then over the base.

The vertical-slice or vertical-cross-section viewpoint is closely related to the disk/washer method and the cylindrical-shell method in calculus, which also compute volume by integrating cross-sectional measures, but with slices oriented along different directions. The choice of vertical versus horizontal slicing depends on which description of the solid is most convenient for the problem at hand.

Examples illustrate the concept: for a prism with a fixed cross-sectional area A0 and a vertical extent Δz, V = A0 Δz. For a base region D with a variable top surface h(x, y), V = ∬_D h(x, y) dA. In such cases, the result depends on the geometry of the base and the variation of height along it.

Notes: the term is not universally used in English-language texts, where the same idea is typically described as volume by vertical cross-sections or by vertical slices. See also volume, cross-sectional area, disk/washer method, and cylinderical-shell method.