Summation of Accumulative Physical Quantity

Besides the extrusion of a filled area profile, volumes of some common solids can also be obtained by expressing an infinitesimal volume element as the revolution of a profile. There are two types of volumes of revolutions used in the calculation of the volumes of some common solids formed by revolution. The difference between the two types of volumes of revolutions is the axis of relvolution relative to the summation direction of the infinitesimal volume elements.

The disc method is the axis of revolution is parallel to the summation direction. This method is simular to the extrusion of filled profile but the filled profile should be defined in the form of πr2 . It is called disc method because the infinitesimal volume elements is in the form of a disc.

The shell method is the axis of revolution is normal to the summation direction or the infinitesimal volume element is summed radially. It is called cylinder method because the infinitesimal volume elements is in the form of a cylindrical shell.

Volumes of Common Solids

1. Solid volume by revolution of profile:

   1. Volume of Cone

      Volume of cone by disc method is same as the summation of a filled profile. Imply

      

      Volume of cone by disc method is obtained by the cross-sectional area of the cone and the height of cone.

      

      Volume of cone by cylindrical shell method is obtained by the cylindrical shell area of the cone and the radius of revolution. Imply

      

      The radius of revolution becomes the variable of integration. Imply

      

   2. Volume of Sphere

      

      Sphere is the special case of ellipsoid in which all three radii are equal.

      The cross-sectional profile at y=0,

      

      The cross-sectional profile at z=0,

      

      The cross-section area

      

      Volume of sphere  by disc method with horizontal summation approach

      

   3. Volume of Oblate Spheroid

      

      Oblate spheroid is the special case of ellipsoid in which two radii are equal and greater than the third radius.

      The cross-sectional profile at y=0,

      

      The cross-sectional profile at z=0,

      

      The cross-section area

      

      Volume of oblate spheroid by disc method with horizontal summation approach

      

   4. Volume of Prolate Spheroid

      

      Prolate spheroid is the special case of ellipsoid in which two radii are equal and less than the third radius.

      The cross-sectional profile at x=0,

      

      The cross-sectional profile at z=0,

      

      The cylindrical shell area

      

      Volume of prolate spheroid by cylindrical shell method with radical summation approach

      

      Volume of prolate spheroid solved through integration by substitution.