Aquaculture Engineering

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…………………a_n .
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+ a_n+ 4(a₂+a₄+a₆+…….+ a_n-1) + 2 (a₃+a₅+a₇+………+a_n-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.