longvolumes

Longvolumes is a term occasionally used to describe the volume of an elongated solid obtained by integrating its cross-sectional area along a fixed longitudinal axis. It is useful when the cross-section, taken perpendicular to the chosen axis, varies with position along that axis.

Mathematically, if a solid occupies positions z in [a, b] and A(z) denotes the area of the cross-section perpendicular to the z-axis at z, the longvolume is V = ∫_a^b A(z) dz. This formulation recovers familiar results: for a constant cross-section A(z)=A, V = A(b−a), the volume of a prism; for a radius that varies with z, the integral yields the volume of a tapered or curved body. For a cylinder (A constant) and a cone or a tapered rod (A(z) changing with z), the same integral framework applies.

Applications appear in engineering and CAD, where designers think in terms of extruded or swept shapes: the volume is computed by integrating cross-sectional area along the extrusion path. The term longvolume is not universally adopted; more common terms include swept volume, extrusion volume, or simply the volume of a solid described by A(z).

See also: volume integral, sweep volume, extruded volume, calculus in three dimensions.