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
    WorkingPrecision Automaticthe 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:

The volume of a standard simplex in 3D:

The volume of a rectangular cuboid:

Volume of the cylinder , expressed in cylindrical coordinates:

Scope  (20)

Special Regions  (7)

Volume of a Cuboid:

Parallelepiped:

Simplex in 3D:

A volume simplex embedded in 4D:

Ball:

Ellipsoid:

Cylinder:

Cone:

Formula Regions  (2)

The volume of a ball represented as an ImplicitRegion:

A cylinder:

The volume of a ball represented as a ParametricRegion:

A cylinder represented with a rational parametrization:

Mesh Regions  (2)

The volume of a MeshRegion:

The volume of a BoundaryMeshRegion:

Derived Regions  (3)

The volume of a RegionIntersection:

The volume of a TransformedRegion:

The volume of a RegionBoundary:

Parametric Formulas  (6)

The volume of an ellipsoid with semimajor axes 3, 2, and 1:

The volume of a hemispherical shell in spherical coordinates:

The volume of a torus of major radius 5 and minor radius 2:

The volume of the product of a disk and a circle embedded in four-dimensional space:

The volume of the paraboloid over the rectangle :

Volume of one octant of a three-sphere using stereographic coordinates:

Options  (3)

Assumptions  (1)

The area of an elliptic pyramid with arbitrary semimajor axis , semiminor axis , and height :

Adding an assumption that the semiaxes are positive simplifies the answer:

The region for , , and :

WorkingPrecision  (2)

Compute the Volume using machine arithmetic:

In some cases, the exact answer cannot be computed:

Find the Volume using 30 digits of precision:

Applications  (6)

A function region :

The region is a volume:

The volume of the region:

Equivalently:

Compute the volume of a polyhedron:

The shape of the Earth is nearly that of an oblate spheroid with volume:

Substitute in the values for the semimajor and semiminor axes:

Find the mass of methanol in a Ball:

Find the mean density of a Cone with a non-uniform mass density defined by :

Compute the volume of empty space in a can with tennis balls, each with a radius of 1.75 inches:

Visualize a can of three balls:

Properties & Relations  (5)

Volume is a non-negative quantity:

Volume[r] is the same as RegionMeasure[r] for 3D regions:

Volume[r] is the same as RegionMeasure[r,3] in general:

Volume[x,s,t,u,c] is equivalent to RegionMeasure[x,{s,t,u},c]:

For a 3D region, Volume is defined as the integral of 1 over that region:

To get the surface volume of a 4D region, use RegionBoundary:

Possible Issues  (2)

The parametric form of Volume computes the volume of possibly multiple coverings:

The region version computes the volume of the image:

The volume of a region of dimension other than 3 is Undefined:

Wolfram Research (2014), Volume, Wolfram Language function, https://reference.wolfram.com/language/ref/Volume.html (updated 2019).

Text

Wolfram Research (2014), Volume, Wolfram Language function, https://reference.wolfram.com/language/ref/Volume.html (updated 2019).

CMS

Wolfram Language. 2014. "Volume." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Volume.html.

APA

Wolfram Language. (2014). Volume. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Volume.html

BibTeX

@misc{reference.wolfram_2022_volume, author="Wolfram Research", title="{Volume}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Volume.html}", note=[Accessed: 08-June-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_volume, organization={Wolfram Research}, title={Volume}, year={2019}, url={https://reference.wolfram.com/language/ref/Volume.html}, note=[Accessed: 08-June-2023 ]}