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

Although triple integration is usually required to determine the volume of 3D body. However volume of 3D body can also be determined by performing a double integration or a single integration.

Volume by Double Integration

 If the inner integration of the unit elemental volume can be expressed as a strip of elemental volume in one dimension.

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

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

An unit elemental volume can be expressed as

Therefore the unit elemental volume can be expressed as a strip of elemental volume of the solid cylinder U in the planar region R of cartesian coordinates yz. Imply

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

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

Starting from horizontal sweeping along y axis

Consider an unit elemental volume ΔVyz along y axis horizontally. 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 can be determined by

Therefore the volume of the solid cone U is

Volume by Single Integration

 If the inner integration of the unit elemental volume can be expressed as a sheet of elemental volume in two dimensions.

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

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

An unit elemental volume can be expressed as

Therefore the unit elemental volume can also be expressed as a sheet of elemental volume of the solid cylinder U along the cartesian coordinate axis x. Imply

Sweeping the unit elemental volume ΔVx  along x axis horizontally

Since all unit elemental volumes of ΔVx are bounded along x axis, imply

Therefore the volume of the solid cone U is

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