# 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 Automatic 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:

RegionMeasure corresponds to curve length for one-dimensional regions:

RegionMeasure corresponds to surface area for two-dimensional regions:

RegionMeasure corresponds to volume for three-dimensional regions:

Area of a bow-tie figure:

Volume of a cylinder expressed in cylindrical coordinates:

## Scope(27)

### Special Regions(10)

The measure for Point corresponds to counts:

Points can be used in any number of dimensions:

The measure for Line corresponds to arc length:

Lines can be used in any number of dimensions:

Rectangle can be used in 2D, and the measure corresponds to area:

Cuboid can be used in any number of dimensions:

A Simplex can correspond to a point, line, or triangle in 2D:

Simplices can be used in any number of dimensions:

The measure of a standard unit simplex in dimension :

Polygon represents an area:

In 3D:

Disk can be used in 2D:

Ball can be used in any dimension, and the measure is the generalized volume:

The measure of unit balls in dimension :

Disk as an ellipse can be used in 2D:

Ellipsoid can be used in any dimension:

Circle can be used in 2D:

Cylinder can be used in 3D:

Cone can be used in 3D:

### Formula Regions(2)

The measure of a disk represented as an ImplicitRegion:

A cylinder volume:

The measure of a disk represented as a ParametricRegion:

Using a rational parametrization of disk:

A cylinder volume:

### Mesh Regions(2)

The measure of a MeshRegion in 2D:

In 3D:

The measure of a BoundaryMeshRegion:

In 3D:

### Derived Regions(3)

The measure of a RegionIntersection:

The measure of a TransformedRegion:

The measure of a RegionBoundary:

### Geographic Regions(2)

The measure of a polygon of geographic entities:

Polygons with GeoPosition:

The measure of a polygon with GeoGridPosition:

### Parametric Formulas(8)

Length of a circular arc:

An infinite curve in polar coordinates with finite length:

The surface area of a torus of major radius 5 and minor radius 2:

The volume of its interior:

The area of a "flat torus" embedded in four-dimensional space:

The hypervolume of a 4-sphere embedded in five dimensions:

The hypervolume of the paraboloidal function graph over the unit hypercube:

The length of a curve "bouncing" between the poles on the unit sphere:

The area of the unit square in stereographic coordinates on the sphere:

## Options(4)

### Assumptions(2)

The implicit region can represent both ellipses and hyperbolas:

Adding the assumption gives the length of an ellipse only:

The area of an ellipse with arbitrary semimajor axes and :

Adding an assumption that the semimajor axes are positive simplifies the answer:

### WorkingPrecision(2)

Compute the arc length using machine arithmetic:

Find the area using 30 digits of precision:

## Applications(13)

### Points(2)

For point sets, the counting measure is used. Each point contributes 1 to the measure:

For constant point mass , multiply the measure by to get the total mass:

For a varying point mass function , use Integrate:

### Curves(4)

The length of a function curve :

The length of an implicit curve:

In 3D:

Find a formula for the length of a Peano curve:

Find the total charge along a wire with constant charge density :

For varying density , use Integrate:

### Surfaces(2)

The area of a function surface :

Total mass for a rectangular region:

With uniform mass density :

With varying mass density given by , use Integrate:

### Solids(3)

Total mass for a Ball with constant density :

For a varying density function , use Integrate:

Find the mass of ethanol in a Cone:

Density of ethanol:

Volume of cone:

Mass of ethanol in the cone:

Find the mass of a Cylinder with a nonuniform mass density defined by :

Density of cylinder:

Volume of cylinder:

Mass of cylinder:

### Higher-Dimensional Regions(2)

Derive a formula for the region measure of an -dimensional unit ball:

The volume of the 3D hypersurface :

## Properties & Relations(10)

RegionMeasure for a region is given by the integral :

ArcLength is a special case of RegionMeasure for one-dimensional regions:

Area is a special case of RegionMeasure for two-dimensional regions:

Volume is a special case of RegionMeasure for three-dimensional regions:

The measure used is determined by RegionDimension, including count for dimension 0:

Length for dimension 1:

Area for dimension 2:

Volume for dimension 3:

For regions containing a mix of dimensions, RegionDimension gives the largest dimension:

Since the dimension is 1, this computes the length:

RegionMeasure[x,{t},c] is equivalent to ArcLength[x,t,c]:

RegionMeasure[x,{s,t},c] is equivalent to Area[x,s,t,c]:

RegionMeasure[x,{s,t,u},c] is equivalent to Volume[x,s,t,u,c]:

RegionCentroid is equivalent to Integrate[p,p]/m with m=RegionMeasure[]:

## Possible Issues(3)

RegionMeasure uses the counting measure for discrete points:

This specifies that the two-dimensional Lebesgue measure should be used:

The parametric form takes the parametrization as fundamental and will count multiple coverings:

The region version computes the measure of the image:

RegionMeasure uses machine arithmetic when the exact answer cannot be computed:

## Neat Examples(1)

Find the measure of the Cantor set:

Compute the measure for the first six iterations:

Find the length for iteration k:

The measure in the limit:

Wolfram Research (2014), RegionMeasure, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionMeasure.html (updated 2019).

#### Text

Wolfram Research (2014), RegionMeasure, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionMeasure.html (updated 2019).

#### CMS

Wolfram Language. 2014. "RegionMeasure." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/RegionMeasure.html.

#### APA

Wolfram Language. (2014). RegionMeasure. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionMeasure.html

#### BibTeX

@misc{reference.wolfram_2021_regionmeasure, author="Wolfram Research", title="{RegionMeasure}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/RegionMeasure.html}", note=[Accessed: 25-June-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2021_regionmeasure, organization={Wolfram Research}, title={RegionMeasure}, year={2019}, url={https://reference.wolfram.com/language/ref/RegionMeasure.html}, note=[Accessed: 25-June-2022 ]}