# RegionMoment

RegionMoment[reg,{i1,i2,,in}]

computes the polynomial moment for the region reg.

# Examples

open allclose all

## Basic Examples(2)

Find the polynomial moment over a ball:

Find same moment with parameters in the region:

Find the moments of different powers:

## Scope(10)

### Special Regions(1)

Find the moment of various special regions in various dimensions:

### Formula Regions(2)

Moment of a disk represented as an ImplicitRegion:

Moment of a cylinder:

The moment of a disk represented as a ParametricRegion:

Using a rational parametrization of a disk:

For a cylinder:

### Mesh Regions(2)

Moment of a MeshRegion in 2D:

In 3D:

The moment of a BoundaryMeshRegion:

In 3D:

### Derived Regions(3)

Moment of a RegionIntersection:

Moment of a TransformedRegion:

The moment of a RegionBoundary:

### Geographic Regions(2)

The moment of a polygon with GeoPosition:

The moment of a polygon with GeoGridPosition:

## Applications(3)

Find the surface area of a cow using the zero-order RegionMoment:

Find its centroid using the first-order moments for each of the axes:

Verify the results using RegionMeasure and RegionCentroid:

Find the covariance matrix of a constant-density distribution defined by a region:

Transform the region so that its RegionCentroid is at the origin:

Compute the matrix of second-order moments and normalize it by dividing by RegionMeasure:

Find a sphere that shares the first four moments with the cow surface:

The sphere has the same surface area and centroid as the cow:

## Properties & Relations(8)

Zero-order moment for curves is equivalent to ArcLength:

Zero-order moment for surfaces is equivalent to Area:

Zero-order moment for volumes is equivalent to Volume:

The zero-order moment for any region is equivalent to the RegionMeasure:

RegionCentroid is the first moments divided by the zero moment:

The centroid is given by :

Compare to the centroid:

MomentOfInertia can compute the moment of inertia matrix wrt to the origin consisting of multiple region moments:

Moment computes for a PDF :

RegionMoment computes corresponding to a uniform density:

CentralMoment computes for a PDF and centroid :

After centering the region, this becomes a standard moment:

As in the previous example, RegionMoment assumes uniform distribution:

Wolfram Research (2016), RegionMoment, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionMoment.html.

#### Text

Wolfram Research (2016), RegionMoment, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionMoment.html.

#### CMS

Wolfram Language. 2016. "RegionMoment." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionMoment.html.

#### APA

Wolfram Language. (2016). RegionMoment. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionMoment.html

#### BibTeX

@misc{reference.wolfram_2024_regionmoment, author="Wolfram Research", title="{RegionMoment}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/RegionMoment.html}", note=[Accessed: 19-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_regionmoment, organization={Wolfram Research}, title={RegionMoment}, year={2016}, url={https://reference.wolfram.com/language/ref/RegionMoment.html}, note=[Accessed: 19-July-2024 ]}