Volume integral

Integral over a 3-D domain In mathematics (particularly multivariable calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities, or to calculate mass from a corresponding density function. In coordinates Often the volume integral is represented in terms of a differential volume element d V = d x d y d z {\displaystyle dV=dx\,dy\,dz} . ∭ D f ( x , y , z ) d V . {\displaystyle \iiint _{D}f(x,y,z)\,dV.} It can also mean a triple integral within a region D ⊂ R 3 {\displaystyle D\subset \mathbb {R} ^{3}} of a function f ( x , y , z ) , {\displaystyle f(x,y,z),} and is usually written as: ∭ D f ( x , y , z ) d x d y d z . {\displaystyle \iiint _{D}f(x,y,z)\,dx\,dy\,dz.} A volume integral in cylindrical coordinates is ∭ D f ( ρ , φ , z ) ρ d ρ d φ d z , {\displaystyle \iiint _{D}f( ho ,\varphi ,z) ho \,d ho \,d\varphi \,dz,} and a volume integral in spherical coordinates (using the ISO convention for angles with φ {\displaystyle \varphi } as the azimuth and θ {\displaystyle \theta } mea