Simpson's Rule

In the previous topic, the Trapezium Rule gave a reasonably accurate method for approximating the area under a curve.

Thomas Simpson used a different approach which led to a more accurate method known as Simpson's Rule.

Instead of using a trapezium, which has straight edges he used a slice of a parabola at the top of each strip. 
He chose the parabola because of the following property of parabolas:

The following formula gives the exact area under any parabola

Area = Y12_Simpsons_Rule_01.gif

 Y12_Simpsons_Rule_02.gif

Because every curve is not a parabola this method does not give an exact answer but it is better than the Trapezium Rule and gives a very accurate answer even for a small number of strips.

There has to be an even number of strips, which means an odd number of values on the x-axis.


Simpson's Rule

In general terms, Simpson's Rule is:

Y12_Simpsons_Rule_03.gif

 

Y12_Simpsons_Rule_04.gif

 

In words

Area = 0ne third x width of each strip x [first ordinate + last ordinate +4(sum of even ordinates)+ 2(sum of odd ordinates)]

(An ordinate is a y-value. This formula is provided for the Bursary examination.)

Example
Solution

a. Complete the table below for the values of y = 15 − x2

x
0
0.5
1
1.5
2
2.5
3
y
              

Substituting the x-values into y = 15 − x2

x
0
0.5
1
1.5
2
2.5
 3
y
15
14.75
14
12.75
11
8.75
 6
b. Use the Simpson's Rule and the table of values to find the approximate area between the curve y = 15 − x2, the x-axis, the y-axis and the line x = 3.

The required area = Y12_Simpsons_Rule_05.gif

The width of each trapezium is 0.5

Area = 1/ 3 x 0.5 x [15 + 6 +4(14.75+12.75+8.75)+2(14+11)

         = 36

The required area is 36 units2

 

Download an interactive spreadsheet (Microsoft Excel) showing the Trapezium Rule and Simpson's Rule.

button_download.gif

(Windows users, right click and "Save target as..." to save the files on your computer.)