Centroid of 2D Plane Body

The centroid of an plate is determined by the first moment of a two dimensional plane body with the method of the first moment of area.

And the centroid of a wire is determined by the first moment of a two dimensional plane body with the method of the first moment of line.

Centroids of Areas

Centroid by Single Integration

The unit elemental areas of an object used to determine the centroid of a 2D plane area can be rearranged into groupped elemental areas. Imply

After the grouping of unit elemental areas into one elemental area, the coordinates of the centroid of an area can also be determined by one single integration in a  similar way by considering the centroid of each elemental area strip. Imply

Centroid of Area by Single Integration

And for curves in polar form

For example, the signed area of the planar region R is bounded by curves in polar form, Imply

The unit element area of a region can be grouped into either a thin circular arc strip or  a thin slice of circular sector. And the element area  GA in polar form becomes

By using a thin circular arc strip, or using a thin slice of circular sector as the element area, the centroid of the planar region can be determined by a single integration through sweeping the the elemental centroid of the elemental area strip or slice along either polar variables. Imply

By sweeping the centroid of circular sector slice along variable angle θ circularly

Centroid of circular sector slice. Imply

Therefore, centroid of the bounded area is

Similarly,

By sweeping the centroid of circular arc strip along variable radius r radically

Centroid of circular arc strip. Imply

Therefore, centroid of the bounded area is

Similarly,