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.

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
$E =$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 $7.75$, $10.70$, $11.20$, $9.70$, $7.75$, $6.80$, $6.30$, $6.80$ and $2.00$ respectively. The width of each strip is $8.25$m.

Solution:
Since the number of ordinates is odd, therefore we use Simpson’s Rule
Area $= \frac{S}{3}\left[ {A + 2D + 4E} \right]$
Here    $S = 8.25$m
$A = 7.75 + 2.00 = 9.75$
$D = 11.20 + 7.75 + 6.30 = 25.25$
$E = 10.70 + 9.70 + 6.80 + 6.80 = 34$
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 $12$m 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 $7.1$m. Find its area.

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