totalvolume

Totalvolume is a term used to denote the measure of three‑dimensional space occupied by a material, object, or region. It is a scalar quantity expressed in cubic units (such as cubic meters or liters) and represents the capacity or extent of space that a body occupies or can contain. In practical contexts, totalvolume often refers to the complete, non-overlapping three‑dimensional extent of an object or a defined region.

In mathematics and geometry, the totalvolume of a region R in three‑dimensional space is defined by the triple integral V = ∭_R dV, where dV denotes an infinitesimal volume element. If a region lies above a base D in the xy-plane and between z = f(x,y) and z = g(x,y), its volume is V = ∬_D [f(x,y) − g(x,y)] dA. For solids of revolution, common formulas include V = π ∫_a^b [R(x)]^2 dx for washers, and V = 2π ∫_a^b x f(x) dx for shells. Simple shapes have explicit formulas: a rectangular prism with sides a, b, c has V = abc; a cylinder with radius r and height h has V = π r^2 h; a sphere of radius r has V = 4/3 π r^3.

Methods for irregular objects include decomposing the object into simpler parts, applying triple integrals, or using numerical approaches such as Monte Carlo integration. Totalvolume is related to, but distinct from, related concepts like density, mass, and capacity, where mass equals density times volume and capacity refers to the maximum volume that a container can hold. See also volume, space, and measurement.