Volumenintegralen

Volumenintegralen, or volume integrals, are triple integrals used to aggregate a quantity over a three‑dimensional region. In mathematical analysis, the volume integral of a function f(x,y,z) over a region D ⊂ R^3 is written as ∭_D f(x,y,z) dV, where dV denotes the volume element. If f is identically 1, the integral equals the volume of D. The choice of coordinate system affects the form of dV and can simplify the region or the integrand. In Cartesian coordinates, dV = dx dy dz; in cylindrical coordinates, dV = r dr dθ dz; in spherical coordinates, dV = ρ^2 sinφ dρ dφ dθ.

Volume integrals are evaluated as iterated integrals using Fubini’s theorem: ∭_D f dV = ∫∫∫ f dξ dη dζ, integrating with respect to one variable at a time once the region D is described by appropriate bounds. Regions are often described by inequalities; changing the order of integration or the coordinate system may be advantageous. Transformations between coordinate systems use Jacobians to preserve volume.

Applications include computing physical quantities for solids with nonuniform density ρ(x,y,z): mass M = ∭_D ρ dV; centers of mass and moments of inertia; total electric charge or probability mass within a region. Volume integrals also underpin calculations in probability, fluid dynamics, and electromagnetism.

Example: the volume of the unit ball is ∭_{ρ≤1} dV = ∫_{0}^{2π} ∫_{0}^{π} ∫_{0}^{1} ρ^2 sinφ dρ dφ dθ = 4/3 π. In practice, choosing coordinates that exploit symmetry often simplifies evaluation.