# 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:
•  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 Infinity 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:

 In:= Out= In:= Out= The volume of a standard simplex in 3D:

 In:= Out= In:= Out= The volume of a rectangular cuboid:

 In:= Out= Volume of the cylinder , expressed in cylindrical coordinates:

 In:= Out= ## Possible Issues(2)

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