Summation of Accumulative Physical Quantity

Volumes of common solid by the summation of a variable filled profile extrusion along one dimension is equal to 

Volumes of Common Solids

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

   1. Volume of Tetrahedron or Triangular Pyramid

      

      The base area of the tetrahedron in terms of base plane edge

      

      The cross-sectional area of a tetrahedron varies with x, imply

      

      Volume of tetrahedron by horizontal summation approach.

      

   2. Volume of Regular Tetrahedron

      Volume of regular tetrahedron is a special case of tetrahedron. The volume of regular tetrahedron can be obtained by letting all edges of the tetrahedron are the same, e.g a.

      

   3. Volume of Retangular Pyramid

      

      Volume of rectangular pyramid by horizontal summation approach.

      

   4. Volume of Square Pyramid

      Volume of square pyramid is a special case of retangular pyramid. The volume of square pyramid can be obtained by letting W=H

      

   5. Volume of Pyramid

      

      Volume of pyramid is a general polyhedron of pyramid with a polygonal base. The volume of a pyramid can be obtained by the base plane area of the polygonal base and the height of the polyhedron.

      

   6. Volume of Cone

      

      Volume of cone can be obtained by the base plane area of the cone base and the height of the cone as in pyramid. Volume of cone by horizontal summation approach.

      

   7. Volume of Octahedron

      

      Volume of octahedron or square bipyramid can be formed from two square pyramids by joining its mirror image about the base of the square pyramid. Therefore the volume of a square bipyramid is equal to the total volume of two square pyramid.  In general, a n-gonal bipyramid is a polyhedron formed by two general polyhedron of pyramid with a polygonal base and the two pyramids are arranged by base to base. Therefore the volume of a bipyramid is equal to the total volume of two pyramids.

      

   8. Volume of Ellipsoid

      

      Volume of ellipsoid can be obtained by the summation of cross-sectional area of the ellipsoid along cross-sectional profile.

      The base plane area normal to x axis is,

      

      The cross-sectional profile at y=0,

      

      The cross-sectional profile at z=0,

      

      Volume of ellipsoid by horizontal summation approach.