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!
Learn more about this topic in the following pages.