Volume

Volume[reg]
gives the volume of the three-dimensional region reg.

Volume[{x1,,xn},{s,smin,smax},{t,tmin,tmax},{u,umin,umax}]
gives the volume of the parametrized region whose Cartesian coordinates are functions of s, t, u.

Volume[{x1,,xn},{s,smin,smax},{t,tmin,tmax},{u,umin,umax},chart]
interprets the as coordinates in the specified coordinate chart.

Details and OptionsDetails and Options

  • A three-dimensional region can be embedded in any dimension greater than or equal to three.
  • In Volume[x,{s,smin,smax},{t,tmin,tmax},{u,umin,umax}], if x is a scalar, Volume returns the volume of the parametric three-region .
  • Coordinate charts in the fifth argument of Volume can be specified as triples in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
  • Volume takes an Assumptions option that can be used to specify assumptions on parameters.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions to make about parameters
    MethodAutomaticmethod to use
  • Specific methods include:
  • Automaticautomatic method selection
    "Integrate"exact symbolic integration
    "NIntegrate"numeric integration
  • Additional method suboptions can be given in the form Method->{,opts}.
  • Any option of Integrate or NIntegrate can be passed as a method suboption to the corresponding method.
  • Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the property of CoordinateChartData.

ExamplesExamplesopen allclose all

Basic Examples  (5)Basic Examples  (5)

The volume of a unit ball in 3D:

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The volume of a standard simplex in 3D:

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The volume of a rectangular cuboid:

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Volume of the cylinder , expressed in cylindrical coordinates:

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The volume of a region of dimension two or lower is :

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The volume of a region of dimension four or higher is :

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Introduced in 2014
(10.0)