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

Area by Integration

Although double integration is usually required to determine the planar area. However a planar area can also be determined by performing a single integration. If the inner integration of the unit elemental area is a thin elemental area.

Area by Single Integration

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

An unit elemental area ΔA in rectangular form can be defined as Δx times Δy. Imply

In general, the unit element area of a region can be extended to either a thin vertical rectangular strip or  a thin horizontal rectangular strip. And the element area ΔA becomes

By using a thin rectangular strip as the element area or applying the method of strip slicing, the signed area of the planar region can be determined by a single integration through  sweeping the signed elemental area strip along either rectangular coordinate axis. Imply

By sweeping the horizontal strip along y axis vertically

By sweeping the vertical strip along x axis horizontally

And for curves in polar form

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

An unit elemental area ΔA in polar form can be approximated by Δr times rΔθ. Imply

In general, the unit element area of a region can be extended to either a thin slice of circular sector or a thin circular arc strip. And the element area ΔA becomes.

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

By sweeping the thin circular sector slice along variable angle θ circularly

By sweeping the thin circular arc strip along variable radius r radically ,