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RegionMoment—Wolfram Language Documentation

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BUILT-IN SYMBOL

RegionMoment[reg,{i₁,i₂,…,i_n}]

computes the polynomial moment for the region reg.

Details and Options Examplesopen 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:

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.htmlBibTeX @misc{reference.wolfram_2025_regionmoment, author="Wolfram Research", title="{RegionMoment}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/RegionMoment.html}", note=[Accessed: 08-July-2025 ]}BibLaTeX @online{reference.wolfram_2025_regionmoment, organization={Wolfram Research}, title={RegionMoment}, year={2016}, url={https://reference.wolfram.com/language/ref/RegionMoment.html}, note=[Accessed: 08-July-2025 ]}

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