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

The using of unit elemental areas of an object to determine the centroid of a 2D plane area can be expressed as

A double integation is needed to evaluate with respect to the two varables. Similar to finding the area of a 2D plane object, the centroid of an area can usually be determined by performing a single integration also.

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 grouped 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

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

The unit element area of a region can be grouped into either a thin vertical rectangular strip or  a thin horizontal rectangular strip. And the elemental area ΔA becomes

Considering the thin rectangular strip as the elemental area, the centroid of the planar region can be determined by a single integration through sweeping the elemental centroid of the elemental area strip along either rectangular coordinate axis accordingly. Imply

By sweeping the centroid of horizontal strip along y axis vertically

Centroid of horizontal strip. Imply

Therefore, centroid of the bounded area is

By sweeping the centroid of vertical strip along x axis horizontally

Centroid of vertical strip. Imply

Therefore, centroid of the bounded area is