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

Area by Double Integration

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

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

In general, the area of a region can be determined by multiple integration through sweeping the signed elemental area starting from along either rectangular coordinate axis. Imply

Starting from horizontal sweeping along x axis

Starting from vertical sweeping along y axis

And for curves in polar form

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

And the curve profile in terms of θ, imply

And other boundary curves are

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

Unlike rectangular form, the polar form of an elemental area ΔA is not a constant but a function of r and in turn a function of θ also.

In general, the area of a region can be determined by multiple integration through sweeping the signed elemental area starting from along either polar variables. Imply

Starting from radical sweeping along variable radius r

Starting from circular sweeping along variable angle θ,