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``` Centroid of 3D Body   Centroids of Volumes   Volume by Integration    Volume by Triple Integration```

# 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

ID: 120600011 Last Updated: 6/21/2012 Revision: 0 Ref:

References

1. I.C. Jong; B.G. rogers, 1991, Engineering Mechanics: Statics and Dynamics
2. F.P. Beer; E.R. Johnston,Jr.; E.R. Eisenberg, 2004, Vector Mechanics for Engineers: Statics

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