Distributed Force
Centroid of 3D Body by Integration
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Centroid of 3D Body
Centroid of 3D BodyThe 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 VolumesVolume by IntegrationVolume by Triple IntegrationFor 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
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ID: 120600011 Last Updated: 2012/6/21 Revision: 0 Ref: 
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