Sideway
output.to from Sideway
Draft for Information Only

Content

Centroid of 3D Body
  Centroids of Volumes
   Volume by Integration
    Volume by Double Integration
    Volume by Single 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.

image

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.

image

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

image

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

image

An unit elemental volume can be expressed as

image

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

image

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

image

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

image

Consider an unit elemental volume ΔVyz along y axis horizontally. Imply

image

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

image

Therefore the volume of the solid cone U can be determined by

image

Therefore the volume of the solid cone U is

image

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.

image

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

image

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

image

An unit elemental volume can be expressed as

image

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

image

Sweeping the unit elemental volume ΔVx  along x axis horizontally

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

image

Therefore the volume of the solid cone U is

image

©sideway

ID: 120600013 Last Updated: 7/3/2012 Revision: 0 Ref:

close

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
close

Latest Updated LinksValid XHTML 1.0 Transitional Valid CSS!Nu Html Checker Firefox53 Chromena IExplorerna
IMAGE

Home 5

Business

Management

HBR 3

Information

Recreation

Hobbies 8

Culture

Chinese 1097

English 339

Reference 79

Computer

Hardware 249

Software

Application 213

Digitization 32

Latex 52

Manim 205

KB 1

Numeric 19

Programming

Web 289

Unicode 504

HTML 66

CSS 65

SVG 46

ASP.NET 270

OS 429

DeskTop 7

Python 72

Knowledge

Mathematics

Formulas 8

Algebra 84

Number Theory 206

Trigonometry 31

Geometry 34

Coordinate Geometry 2

Calculus 67

Complex Analysis 21

Engineering

Tables 8

Mechanical

Mechanics 1

Rigid Bodies

Statics 92

Dynamics 37

Fluid 5

Fluid Kinematics 5

Control

Process Control 1

Acoustics 19

FiniteElement 2

Natural Sciences

Matter 1

Electric 27

Biology 1

Geography 1


Copyright © 2000-2024 Sideway . All rights reserved Disclaimers last modified on 06 September 2019