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Volume

gives the volume of the three-dimensional region reg.

Volume[{x₁,…,x_n},{s,s_min,s_max},{t,t_min,t_max},{u,u_min,u_max}]

gives the volume of the parametrized region whose Cartesian coordinates x_i are functions of s, t, u.

Volume[{x₁,…,x_n},{s,s_min,s_max},{t,t_min,t_max},{u,u_min,u_max},chart]

interprets the x_i 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,s_min,s_max},{t,t_min,t_max},{u,u_min,u_max}], 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:

	AccuracyGoal	Infinity	digits of absolute accuracy sought
	Assumptions	$Assumptions	assumptions to make about parameters
	GenerateConditions	Automatic	whether to generate conditions on parameters
	PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
	PrecisionGoal	Automatic	digits of precision sought
	WorkingPrecision	Automatic	the 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:

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:

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