Aquaculture Engineering

Calculation of volume of irregular surfaces - Simpson’s II

Unit 6- Calculation of area and volume

Calculation of volume
Calculation of volume of irregular surfaces

Simpson’s second 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 .
Simpson’s rule for four evenly spaced ordinates becomes
v₁= 3/8 h (a₁+3a₂+3a₃+a₄)
v₂= 3/8 h (a4+3a₅+3a₆+a₇)
v₃= 3/8 h (a₇+3a₈+3a₉+a₁₀)
v₄= 3/8 h (a₁₀+3a₁₁+3a₁₂+ a₁₃)
In general
A = v₁ + v₂ + v₃ + v₄ + v₅ +.....
A = 3/8 h (a₁+3a₂+3a₃+v₄) +3/8 h (a₄+3a₅+3a₆+a₇) +3/8 h (a₇+3a₈+3a₉+a₁₀) +
3/8 h (a₁₀+3a₁₁+3a₁₂+a₁₃) +.......
A=3/8h (a₁+3a₂+3a₃+2a₄+3a₅+3a₆+2a₇+3a₈+3a₉+2a₁₀+…………..+a_n)
Thus the common multiplier in this case is 3/8 times the common internal “ h” and the individual multipliers 1,3,3,2,3,3,2,3,3,1. It is suitable for 4, 7, 10, 13, 16, 19, 22 etc., ordinates.