## 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 =  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:  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 = 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 