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

      

Summation of Accumulative Physical Quantity

Areas of plane shapes can be determined by integration when the area of a plane shape is expressed as an infinitesimal area element that can be summed by the application of integration. 

Areas of plane shapes 

1. Area of Ellipse

   

   Area of ellipse by area under curve.

   

   By subsitution

   

2. Area of Circle

   Area of circle is a special case of ellipse. The area of circle can be obtained by letting b=a

   

3. Area of Circular Sector

   

   Area of circular sector by horizontal summation approach

   Area of triangles

   

   Area under arc

   

   Area of circular sector

   

4. Area of circular segment

   

   Area of circular segment by area subtraction

   

   Area of circular segment in terms of height of arced portion, h

   

Summation of Accumulative Physical Quantity

Areas of plane shapes can be determined by integration when the area of a plane shape is expressed as an infinitesimal area element that can be summed by the application of integration. 

Areas of plane shapes 

1. Area of Rectangle

   

   Area by integration.

   

2. Area of Square

   Area of square is a special case of rectangle. The area of square can be obtained by letting b=a

   

3. Area of Parallelogram

   

   Area of parallelogram by horizontal summation approach

   

   Area of parallelogram by vertical summation approach

   

4. Area of Triangle

   For right angle triangle

   

   Area of right angle triangle by horizontal summation approach

   

   For oblique triangle

   

   Area of oblique triangle by area between curves

   

5. Area of Rhombus

   

   Area of rhombus by area between curves

   

6. Area of Trapezoid

   

   Area of trapezoid by area between curves