## Simpson's Rule

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. [caption id="attachment_762" align="alignnone" width="598" caption="simpson rule"][/caption]

*...where h is the width of each strip and*The more strips involved or you have in the rule the better the approximation.

**n**is the even number## Example

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**to

**x=3**, 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.

-1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |

-1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |

**6.80**This is how you use the Simpson's rule. It's that simple!

*...where h is the width of each strip and*The more strips involved or you have in the rule the better the approximation.

**n**is the even number## Example

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**to

**x=3**, 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.

-1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |

-1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |

**6.80**This is how you use the Simpson's rule. It's that simple!

#### Objectives

A list of sections

#### Tips

A list of tips