Site pages
Current course
Participants
General
Topic 1
Topic 2
Topic 3
Topic 4
Topic 5
Topic 6
Topic 7
Topic 8
Topic 9
Topic 10
Topic 11
Topic 12
Topic 13
Topic 14
Topic 15
Topic 16
Topic 17
Topic 18
Topic 19
Topic 20
Topic 21
Topic 22
Topic 23
Topic 24
Topic 25
Topic 26
Topic 27
Topic 28
Topic 29
Topic 30
Topic 31
Topic 32
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…………………Yn .
The area within ordinates number 1 and 3 is
a1 = 1/3 h (Y1+4Y2+Y3)
The area within 3 rd and 5th 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+Yn+4(Y2+Y4+Y6+…….+Yn-1) + 2 (Y3+Y5+Y7+………+Yn-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.
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…………………Yn .
The area within ordinates number 1 and 3 is
a1 = 1/3 h (Y1+4Y2+Y3)
The area within 3 rd and 5th 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+Yn+4(Y2+Y4+Y6+…….+Yn-1) + 2 (Y3+Y5+Y7+………+Yn-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.
Last modified: Tuesday, 26 April 2011, 7:19 AM