tripleintegral

A triple integral, or triple integral, is a type of definite integral that integrates a function over a three-dimensional region D in three-dimensional space. It is written as ∭_D f(x, y, z) dV, where f is a function defined on D and dV denotes an infinitesimal volume element. The value of the triple integral represents the accumulated quantity f over the volume D, such as total mass for a density function f, total charge, or other spatial totals.

The region D is a subset of R^3 and is often described by inequalities or surfaces. D can be a simple box-like region in Cartesian coordinates, or a more complex region bounded by surfaces. The choice of coordinate system and description of D influence the ease of evaluation.

Evaluation is typically performed as an iterated integral, with the order of integration chosen to simplify the computation. The integral is written as ∭_D f(x, y, z) dx dy dz, but the differentials may be arranged in any order corresponding to the chosen coordinates. Fubini’s theorem allows changing the order of integration when the integrand is integrable, enabling flexibility in setup and calculation.

Common coordinate systems include:

- Cartesian coordinates: dV = dx dy dz

- Cylindrical coordinates: dV = r dr dθ dz

- Spherical coordinates: dV = ρ^2 sin φ dρ dφ dθ

Applications of triple integrals include computing volumes, masses with a density function, moments of inertia, and other physical properties. They are related to other theorems in vector calculus, such as the divergence theorem, which connects volume integrals to surface integrals.

Example: the volume of a sphere of radius R is ∭_{x^2+y^2+z^2≤R^2} 1 dV, yielding V = 4/3 π R^3.