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
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.
In general terms, Simpson's Rule is:
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.)
a. Complete the table below for the values of y = 15 − x2
Substituting the x-values into y = 15 − x2
|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)
The required area is 36 units2
Download an interactive spreadsheet (Microsoft Excel) showing the Trapezium Rule and Simpson's Rule.
(Windows users, right click and "Save target as..." to save the files on your computer.)