Simpson's Rule

The most important rule in practice is Simpson’s Rule because of its simplicity and accuracy. When more accuracy is required, this rule should be used.

To find the area ABCD as shown in the figure, the base AD must be divided into an even number of strips of equal width S, producing an odd number of ordinates. The length of each ordinate a,b,c,d,e,f,g is accurately measured. The first and the last ordinates are called the extreme ordinates; the second, fourth, sixth, etc., the even ordinates; and the third, fifth, seventh, etc. are the odd ordinates.


simpsons-rules

            By Simpson’s Rule, the area is determined as:
Area  = \frac{S}{3}\left[ {A + 2D + 4E} \right]
Where A = sum of the first and the last ordinate
B = sum of the odd ordinates
C = sum of the even ordinates
S = width of each strip

Note: This rule is applicable only when there is an even number of strips or odd number of ordinates.

 

Example:

Find the area of an irregular figure whose ordinates are 7.75, 10.70, 9.70, 7.75, 6.80, 6.30, 6.80 and 2.00 respectively. The width of each strip is 8.25m.

Solution:
            Since the number of ordinates is odd, therefore we use Simpson’s Rule
Area  = \frac{S}{3}\left[ {A + 2D + 4E} \right]
Here    S = 8.25m
A = 7.75 + 2.00 = 9.75
D = 11.20 + 7.75 + 6.30 = 25.25
E = 10.70 + 9.70 + 6.80 + 6.80 = 34
Area  = \frac{{8.25}}{3}\left[ {9.75 + 2\left( {25.25} \right) + 4\left( {34} \right)} \right]
 = 2.75\left( {196.25} \right) = 53.96 square meters.

 

Example:

A parabolic piece of cardboard from root to tip is 12m long. At 9 equidistant places, the widths are: 0.0, 1.3, 3.7, 4.0, 4.8, 5.8, 6.2, 6.9 and 7.1m. Find its area.

Solution:
            Since the Simpson’s Rule is
Area  = \frac{S}{3}\left[ {A + 2D + 4E} \right]
Here    S = 12m
A = 0.0 + 7.1 = 7.1
D = 3.7 + 4.8 + 6.2 = 14.7
E = 1.3 + 4.0 + 5.8 + 6.9 = 18.0
Area  = \frac{{12}}{3}\left[ {7.1 + 2\left( {14.7} \right) + 4\left( {18} \right)} \right]
 = 4\left[ {7.1 + 29.4 + 72} \right] = 434 square meters.