# 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:
•  Assumptions \$Assumptions assumptions to make about parameters Method Automatic method to use WorkingPrecision Infinity the precision used in internal computations
• Specific methods include:
•  Automatic automatic 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 "ParameterRangeAssumptions" property of CoordinateChartData.

# Examples

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## 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 0:

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

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