Volume integral

Template:Short description {{#invoke:sidebar|collapsible | class = plainlist | titlestyle = padding-bottom:0.25em; | pretitle = Part of a series of articles about | title = Calculus | image = $\underset{a}{\overset{b}{\int }}{f}^{\prime }(t)dt=f(b)-f(a)$ | listtitlestyle = text-align:center; | liststyle = border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa; | expanded = multivariable | abovestyle = padding:0.15em 0.25em 0.3em;font-weight:normal; | above =

• Fundamental theorem

• Limits
• Continuity

• Rolle's theorem
• Mean value theorem
• Inverse function theorem

| list2name = differential | list2titlestyle = display:block;margin-top:0.65em; | list2title = Template:Bigger | list2 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | heading1 = Definitions
 | content1 =

• Derivative (generalizations)
• Differential
  • infinitesimal
  • of a function
  • total

 | heading2 = Concepts
 | content2 =

• Differentiation notation
• Second derivative
• Implicit differentiation
• Logarithmic differentiation
• Related rates
• Taylor's theorem

 | heading3 = Rules and identities
 | content3 =

• Sum
• Product
• Chain
• Power
• Quotient
• L'Hôpital's rule
• Inverse
• General Leibniz
• Faà di Bruno's formula
• Reynolds

}}

| list3name = integral | list3title = Template:Bigger | list3 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

• Lists of integrals
• Integral transform
• Leibniz integral rule

| heading2 = Definitions

 | content2 =

• Antiderivative
• Integral (improper)
• Riemann integral
• Lebesgue integration
• Contour integration
• Integral of inverse functions

 | heading3 = Integration by
 | content3 =

• Parts
• Discs
• Cylindrical shells
• Substitution (trigonometric, tangent half-angle, Euler)
• Euler's formula
• Partial fractions (Heaviside's method)
• Changing order
• Reduction formulae
• Differentiating under the integral sign
• Risch algorithm

}}

| list4name = series | list4title = Template:Bigger | list4 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

• Geometric (arithmetico-geometric)
• Harmonic
• Alternating
• Power
• Binomial
• Taylor

 | heading2 = Convergence tests
 | content2 =

• Summand limit (term test)
• Ratio
• Root
• Integral
• Direct comparison
• 
  Limit comparison
• Alternating series
• Cauchy condensation
• Dirichlet
• Abel

}}

| list5name = vector | list5title = Template:Bigger | list5 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

• Gradient
• Divergence
• Curl
• Laplacian
• Directional derivative
• Identities

 | heading2 = Theorems
 | content2 =

• Gradient
• Green's
• Stokes'
• Divergence
• generalized Stokes
• Helmholtz decomposition

}}

| list6name = multivariable | list6title = Template:Bigger | list6 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | heading1 = Formalisms
 | content1 =

• Matrix
• Tensor
• Exterior
• Geometric

 | heading2 = Definitions
 | content2 =

• Partial derivative
• Multiple integral
• Line integral
• Surface integral
• Volume integral
• Jacobian
• Hessian

}}

| list7name = advanced | list7title = Template:Bigger | list7 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

• Calculus on Euclidean space
• Generalized functions
• Limit of distributions

}}

| list8name = specialized | list8title = Template:Bigger | list8 =

• Fractional
• Malliavin
• Stochastic
• Variations

| list9name = miscellanea | list9title = Template:Bigger | list9 =

• Precalculus
• History
• Glossary
• List of topics
• Integration Bee
• Mathematical analysis
• Nonstandard analysis

}}

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

It can also mean a triple integral within a region $D\subset {ℝ}^3$ of a function $f(x,y,z),$ and is usually written as: $$\underset{D}{\iiint }f(x,y,z)dxdydz.$$

A volume integral in cylindrical coordinates is $$\underset{D}{\iiint }f(\rho ,\phi ,z)\rho d\rho d\phi dz,$$ and a volume integral in spherical coordinates (using the ISO convention for angles with $\phi$ as the azimuth and $\theta$ measured from the polar axis (see more on conventions)) has the form $$\underset{D}{\iiint }f(r,\theta ,\phi )r^2\sin \theta drd\theta d\phi .$$

Example

Integrating the equation $f(x,y,z)=1$ over a unit cube yields the following result: $$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}1dxdydz=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}(1-0)dydz=\underset{0}{\overset{1}{\int }}(1-0)dz=1-0=1$$

So the volume of the unit cube is 1 as expected. This is rather trivial however, and a volume integral is far more powerful. For instance if we have a scalar density function on the unit cube then the volume integral will give the total mass of the cube. For example for density function: $$\{\begin{array}{c}f:{ℝ}^3\to ℝ\\ f:(x,y,z)\mapsto x+y+z\end{array}$$ the total mass of the cube is: $$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}(x+y+z)dxdydz=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}(\frac{1}{2}+y+z)dydz=\underset{0}{\overset{1}{\int }}(1+z)dz=\frac{3}{2}$$

See also

• Divergence theorem
• Surface integral
• Volume element
• Line element
• Line integral

External links

• Template:Springer
• Template:MathWorld