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 OptionsDetails 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$Assumptionsassumptions to make about parameters
    MethodAutomaticmethod to use
    WorkingPrecisionInfinitythe precision used in internal computations
  • Specific methods include:
  • Automaticautomatic 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.

ExamplesExamplesopen allclose all

Basic Examples  (6)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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Introduced in 2014
(10.0)
| Updated in 2016
(10.4)
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