Aquaculture Engineering

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

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 I st rule

Let us consider a curvilinear figure; it can be divided into number of small strips by covering it with “n” equally spaced ordinates, which are at a distance “h” apart. The breadth of the ordinates are in the order of Y₁, Y₂, Y₃, Y₄…………………Y_n .
The area within ordinates number 1 and 3 is
a₁ = 1/3 h (Y₁+4Y₂+Y₃)
The area within 3 ^rd and 5^th ordinates
a₂ = 1/3 h (Y₃+ 4Y₄+Y₅)
The area within 5 th and 7th ordinates
a₃ = 1/3 h (Y₅+4Y₆+Y₇)
The total area is given by
A = a₁ + a₂ + a₃ + a₄ + a₅ +......
A = 1/3 h (Y₁+4Y₂+Y₃) +1/3 h (Y₃+ 4Y₄+Y₅) + 1/3 h (Y₅+4Y₆+Y₇) + - - -
A= h/3 (Y₁+4Y₂+2Y₃+4Y₄+2Y₅+4Y₆+2Y₇) + - - -
A= h/3 [ (Y₁+Y_n+4(Y₂+Y₄+Y₆+…….+Y_n-1) + 2 (Y₃+Y₅+Y₇+………+Y_n-2) ]
This is the generalized form of the Simpson’s first rule applied to areas. 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., ordinates.