Find the polynomial moment over a ball:

Find same moment with parameters in the region:

Find the moments of different powers:

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:

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: