# RegionMeasure

RegionMeasure[reg]

gives the measure of the region reg.

RegionMeasure[reg,d]

gives the d-dimensional measure of the region reg.

RegionMeasure[{x1,,xn},{{t1,a1,b1},,{tk,ak,bk}}]

gives the k-measure of the parametric formula whose Cartesian coordinates xi are functions of tj.

RegionMeasure[{x1,,xn},{{t1,a1,b1},,{tk,ak,bk}},chart]

interprets the xi as coordinates in the specified coordinate chart.

# Details and Options

• RegionMeasure is also known as count (0D), length (1D), area (2D), volume (3D), and Lebesgue measure.
• Example cases where rows correspond to embedding dimension and columns to geometric dimension:
• If the region reg is of dimension d0, then the d-dimensional measure is used.
• The zero-dimensional measure counts the number of points in the region.
• In RegionMeasure[x,{{t1,a1,b1},,{tk,ak,bk}}], if x is a scalar, RegionMeasure returns the measure of the hypersurface {t1,,tk,x} in k+1 dimensions.
• Coordinate charts in the third argument of RegionMeasure 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(6)

RegionMeasure corresponds to count for zero-dimensional regions:

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RegionMeasure corresponds to curve length for one-dimensional regions:

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RegionMeasure corresponds to surface area for two-dimensional regions:

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RegionMeasure corresponds to volume for three-dimensional regions:

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Area of a bow-tie figure:

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Volume of a cylinder expression in cylindrical coordinates:

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