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

Volume by Integration

Volume by Triple Integration

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

An elemental volume ΔV in rectangular form can be defined as Δx times Δy times Gz. Imply

Therefore the volume of the solid cone U in cartesian coordinates xyz is equal to

In general, the volume of a region can be determined by multiple integration through sweeping the signed elemental volume starting from along any one of the rectangular coordinate axes.  Imply

Starting from horizontal sweeping along x axis

Considering an elemental volume along x axis.  Imply

All elemental volumes can be bounded by curves in the plane yz. And the curves is

Similarly sweeping the elemental volume ΔVyz along y axis horizontally.

Considering an elemental volume ΔVz  along y axis.  Imply

Since the bounding curves are joined at plane zx, The bounds of the bounding curves are

Therefore the volume of the solid cone U is

The volume of the solid cone U can also be determined starting from other axis.

Starting from horizontal sweeping along y axis

Considering an elemental volume along y axis.  Imply

All elemental volumes in z direction can be bounded by curves in the plane zx. And the curves is

Similarly sweeping the elemental volume ΔVzx along z axis vertically.

Considering an elemental volume ΔVx  along z axis.  Imply

The bounding curves in x direction can also be bounded at plane zx. The bounds of the bounding curves are

The volume of the solid cone U can be expressed as

Therefore the volume of the solid cone U is