Centroid of 3D Body

The centroid of 3D Body is determined by the first moment of a three dimensional body with the method of the first moment of volume.

Centroids of Volumes

The using of unit elemental volume of an object to determine the centroid of a 3D body volume can be expressed as

Centroid by Integration

A triple integation is needed to evaluate with respect to the three varables. Similar to finding the volume of a 3D object body, the centroid of a 3D volume can usually be determined by performing a single integration or a double integration also.

Centroid by Double Integration

The unit elemental volumes of an object used to determine the centroid of a 3D body can be rearranged into grouped elemental volumes. Imply

After the grouping of unit elemental volumes into one elemental volume, the coordinates of the centroid of a volume can also be determined by double integration in a  similar way by considering the centroid of each elemental volume block. Imply

Centroid of Volume by Double Integration

For example, the signed volume of the 3D ellipic cylinder is bounded by surfaces in rectangular form , Imply

The unit element volume of a region can be grouped into either a small vertical rectangular block or  a small horizontal rectangular block . And the elemental volume ΔV becomes

Considering the small horizontal rectangular block as the element volume, the centroid of the rectangular block can be determined by a single integration through sweeping the elemental centroid of the elemental rectangular block along the rectangular coordinate axis accordingly. Imply

By sweeping the element of horizontal rectangular block along x axis horizontally

Centroid of horizontal rectangular block. Imply

Therefore, centroid of the bounded volume is

Centroid by Single Integration

The unit elemental volumes of an object used to determine the centroid of a 3D body can be rearranged into grouped elemental volumes. Imply

After the grouping of unit elemental volumes into one elemental volume, the coordinates of the centroid of a volume can also be determined by one single integration in a  similar way by considering the centroid of each elemental volume block. Imply

Centroid of Volume by Double Integration

Volume by Single Integration

For example, the signed volume of the 3D ellipic cylinder is bounded by surfaces in rectangular form , Imply

The unit element volume of a region can be grouped into either a thin vertical plane sheet or  a thin horizontal plane sheet . And the elemental volume ΔV becomes

Considering the thin vertical plane sheet as the element volume, the centroid of the thin vertical plane sheet can be determined by a double integration through sweeping the elemental centroid of the elemental plane sheet along the rectangular coordinate axis accordingly. Imply

By staring with sweeping the element of vertical plane sheet along y axis horizontally

Centroid of vertical plane sheet. Imply

Therefore, centroid of the bounded volume is