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Centroid - Integral formula

Centroid - Integral formula: Encyclopedia II - Centroid - Integral formula

The abscissa of the centroid of a plane figure can be given as the integral , where f(x) is the vertical extent of the object at abscissa x. The same formula yields the first coordinate of the centroid of an object in , for any dimension n, provided that f(x) is the (n − 1)-dimensional measure of the object's cross-section at coordinate See also:

Centroid, Centroid - Centroid of triangle and tetrahedon, Centroid - Centroids of cones and pyramids, Centroid - Centroid and convexity, Centroid - Integral formula, Centroid - Center of symmetry, Centroid - Physical centroids

Centroid, Centroid - Center of symmetry, Centroid - Centroid and convexity, Centroid - Centroid of triangle and tetrahedon, Centroid - Centroids of cones and pyramids, Centroid - Integral formula, Centroid - Physical centroids, Pappus's centroid theorem

Centroid: Encyclopedia II - Centroid - Integral formula

Centroid - Integral formula

The abscissa of the centroid of a plane figure can be given as the integral , where f(x) is the vertical extent of the object at abscissa x.

The same formula yields the first coordinate of the centroid of an object in , for any dimension n, provided that f(x) is the (n − 1)-dimensional measure of the object's cross-section at coordinate x — that is, the set of all points in the object whose first coordinate is x.

Note that the denominator is simply the object's n-dimensional measure. The formula cannot be applied if the object has zero measure, or if either integral diverges.