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Calculation of volume of irregular surfaces - Simpson’s I
Unit 6- Calculation of area and volume
Calculation of volume
Calculation of volume of irregular surfaces
Simpson’s I st rule
Let us consider a curvilinear figure. It can be divided into number of small areas by covering it with “n” equally spaced parts, which are at a distance “h” apart. The areas are in order being a1, a2, a3, a4…………………an .
The volume within ordinates number 1 and 3 is
v1 = 1/3 h (a1+4a2+a3)
The volume within 3 rd and 5th ordinates
v2 = 1/3 h (a3+ 4a4+a5)
The volume within 5 th and 7th ordinates
v3 = 1/3 h (a5+4a6+aY7)
The total volume is given by
V = v1 + v2 + v3 + v4 + v5 +......
V = 1/3 h (a1+4a2+a3) +1/3 h (a3+ 4a4+a5) + 1/3 h (a5+4a6 +a7) +......
V= h/3 (a1+4a2+2a3+4a4+2a5+4a6+a7) +.....
V= h/3 [(a1+ an+ 4(a2+a4+a6+…….+ an-1) + 2 (a3+a5+a7+………+an-2) ]
This is the generalized form of the Simpson’s first rule applied to volume. The common multiplier is 1/3 x the common interval “h” and the individual multipliers are 1,4,2,4,2,4,2,4,2,4,1. It is suitable for 3, 5, 7, 9, 11, 13 etc., areas.
The volume within ordinates number 1 and 3 is
v1 = 1/3 h (a1+4a2+a3)
The volume within 3 rd and 5th ordinates
v2 = 1/3 h (a3+ 4a4+a5)
The volume within 5 th and 7th ordinates
v3 = 1/3 h (a5+4a6+aY7)
The total volume is given by
V = v1 + v2 + v3 + v4 + v5 +......
V = 1/3 h (a1+4a2+a3) +1/3 h (a3+ 4a4+a5) + 1/3 h (a5+4a6 +a7) +......
V= h/3 (a1+4a2+2a3+4a4+2a5+4a6+a7) +.....
V= h/3 [(a1+ an+ 4(a2+a4+a6+…….+ an-1) + 2 (a3+a5+a7+………+an-2) ]
This is the generalized form of the Simpson’s first rule applied to volume. The common multiplier is 1/3 x the common interval “h” and the individual multipliers are 1,4,2,4,2,4,2,4,2,4,1. It is suitable for 3, 5, 7, 9, 11, 13 etc., areas.
Last modified: Tuesday, 26 April 2011, 10:00 AM