Before attempting this reading it would be a good idea to learn the trapezium rule here
, because I won’t go so much into details about the logic behind. I will concentrate more on how to use the rule rather than how it works. Thomas Simpson (20 August 1710 – 14 May 1761) was a British mathematician, inventor and eponym of Simpson’s rule to approximate definite integrals.Suppose we wanted to find the area under the graph of y = f(x)
between x = a
and x = b
The graph for the problem is shown below.
We would usually integrate to find out the area for this so the area would be;
There might come a situation when we can’t integrate the function in this case we do an approximation. The Simpson’s rule is very logical. The following diagram shows a graph followed by the Simpson’s rule.
…where h is the width of each strip and n is the even number
The more strips involved or you have in the rule the better the approximation.
Use the Simpson’s rule with 8 strips to calculate an approximation to the following integral:
First we need to work out the width of each strip to use for h
in the Simpon’s rule. We know that the region we want to find the area for runs from x=-1
, so the total width is a 4
. That must mean that the width of each strip is 4/8=0.5
So we now need to work out the y
values at each point. A good way of finding these values is by drawing a table as below.For example: when x = -1
This is how we find the x values.
So we now since we know the y values at each x point we can move on to using the Simpson’s rule as the following;
And the answer for the area between the specified points is 6.80
This is how you use the Simpson’s rule. It’s that simple!