Aquaculture Engineering (2+1)

Calculation of area of irregular plane surfaces - Simpson’s II

Unit 6- Calculation of area and volume

Calculation of area of irregular plane surfaces

Calculation of area of irregular surfaces can be calculated using the following formulae.

Simpson's second rule

Let us consider a curvilinear figure. It can be divided into number of strips by covering it with “n” equally spaced ordinates, which are at a distance “h” apart. The breadth of the ordinates in the order of Y₁, Y₂, Y₃, Y₄…………………Y_n .
Simpson’s second rule for four evenly spaced ordinates becomes
a₁= 3/8 h (Y₁+3Y₂+3Y₃+Y₄)
a₂= 3/8 h (Y₄+3Y₅+3Y₆+Y₇)
a₃= 3/8 h (Y₇+3Y₈+3Y₉+Y₁₀)
a₄= 3/8 h (Y₁₀+3Y₁₁+3Y₁₂+Y₁₃)
In general
A = a₁ + a₂ + a₃ + a₄ + a₅ + ........
A = 3/8 h (Y₁+3Y₂+3Y₃+Y₄) +3/8 h (Y₄+3Y₅+3Y₆+Y₇) +3/8 h (Y₇+3Y₈+3Y₉+Y₁₀) +
3/8 h (Y₁₀+3Y₁₁+3Y₁₂+Y₁₃) + .....
A=3/8h (Y₁+3Y₂+3Y₃+2Y₄+3Y₅+3Y₆+2Y₇+3Y₈+3Y₉+2Y₁₀+……………………..+Y_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.