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 xi are functions of s, t, u.

Volume[{x1,,xn},{s,smin,smax},{t,tmin,tmax},{u,umin,umax},chart]

interprets the xi as coordinates in the specified coordinate chart.

Details 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 {s,t,u,x}.
  • Coordinate charts in the fifth argument of Volume can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
  • The following options can be given:
  • AccuracyGoalInfinitydigits of absolute accuracy sought
    Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate conditions on parameters
    PerformanceGoal$PerformanceGoalaspects of performance to try to optimize
    PrecisionGoalAutomaticdigits of precision sought
    WorkingPrecisionInfinitythe precision used in internal computations
  • Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the "ParameterRangeAssumptions" property of CoordinateChartData.

Examples

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Basic Examples  (4)

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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Scope  (20)

Options  (3)

Applications  (6)

Properties & Relations  (5)

Possible Issues  (2)

Introduced in 2014
(10.0)
|
Updated in 2019
(12.0)