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Summation of Accumulative Physical Quantity
  Volumes of Common Solids

Summation of Accumulative Physical Quantity

Volumes of common solid can be determined by integration when the volume of a solid is expressed as an infinitesimal volume element that can be summed by the application of integration. 

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The infinitesimal volume element can be formed by the extrusion of a filled profile or the revolution of a profile.

Volumes of Common Solids

  1.  Solid volume by the extrusion of constant filled profile:

    1. Volume of Cuboid

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      Volume of cuboid by horizontal summation approach.

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    2. Volume of Cube

      Volume of cube is a special case of cuboid. The volume of cube can be obtained by letting W=H=L

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    3. Volume of Cylinder

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      Volume of cylinder by horizontal summation approach

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    4. Volume of Triangular Shaped Prism

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      Volume of triangular shaped prism by horizontal summation approach

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    5. Volume of Parallelepiped

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      A parallelepiped is a solid that all sides are parallelograms. A parallelepiped has a constant cross-sectional area along its height. Although the cross-sectional area is a function of y and z, the total cross-sectional area s a constant along x axis, therefore volume of parallelepiped can be obtained by horizontal summation approach.

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      The cross-section area of a parallelepiped is

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      The height, edge, angles and other construction curves of the parallelepiped is

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      The length of curve lc is

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      The height of parallelepiped is

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      Therefore the volume of parallelepiped is

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ID: 111100015 Last Updated: 12/4/2011 Revision: 1 Ref:

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References

  1. S. James, 1999, Calculus
  2. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
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