# 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:
•  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.
• RegionMeasure can be used with symbolic regions in GeometricScene.

# Examples

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## Basic Examples(6)

RegionMeasure corresponds to count for zero-dimensional regions:

 In:= In:= Out= In:= Out= RegionMeasure corresponds to curve length for one-dimensional regions:

 In:= In:= Out= In:= Out= In:= Out= RegionMeasure corresponds to surface area for two-dimensional regions:

 In:= In:= Out= In:= Out= In:= Out= RegionMeasure corresponds to volume for three-dimensional regions:

 In:= In:= Out= In:= Out= In:= Out= Area of a bow-tie figure:

 In:= Out= In:= Out= Volume of a cylinder expressed in cylindrical coordinates:

 In:= Out= ## Neat Examples(1)

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