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. 

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

      

      Volume of cuboid by horizontal summation approach.

      

   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

      

   3. Volume of Cylinder

      

      Volume of cylinder by horizontal summation approach

      

   4. Volume of Triangular Shaped Prism

      

      Volume of triangular shaped prism by horizontal summation approach

      

   5. Volume of Parallelepiped

      

      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.

      

      The cross-section area of a parallelepiped is

      

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

      

      The length of curve lc is

      

      The height of parallelepiped is

      

      Therefore the volume of parallelepiped is