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### Simpson's Rule - Alternating Pattern

```
Date: 09/20/97 at 14:49:53
From: Keith Holm
Subject: Simpson's rule

I'm a computer science student who is trying to apply Simpson's rule
to a program, but I do not understand it well enough to proceed.  Can
somebody give a 10 minute "this is how it works"?  The book I have
shows a series from low to high bounds, but does not explain the
alternating 4F 2F pattern.
```

```
Date: 09/20/97 at 16:13:23
From: Doctor Anthony
Subject: Re: Simpson's rule

I will give you the formula to apply without proof.  If you want to
see the derivation of the formula, write back and I will go through it
in detail.

The formula is exact for polynomials up to cubics, so if for some
strange reason you wanted to use Simpson's rule to integrate
3x^3 - x^2 -5x -3 then just two equal intervals as wide as you please
could be used. In the general case we divide the the total interval
into an EVEN number of equal sub-intervals, and the more of these,
the more accurate will be the final answer. If you have just two
sub-intervals, there will be three ordinates y0, y1 and y2 to be
calculated. If you have 4 sub-intervals, then you would have 5
ordinates y0, y1,...., y4  to calculate.

It is important to use a standard system for naming the ordinates,
e.g. y2 is an even-numbered ordinate, y3 is an odd-numbered ordinate,
and the formula requires that you treat these in different ways.
So always start with y0; then the formula will work correctly in the
form I give it to you.

Suppose you divide the total range into n intervals, where n MUST be
an even number. This means, of course, that there will be an odd
number of ordinates. The first ordinate will be y0; the last ordinate
will be yn. The width of the sub-intervals is h.

The area under the curve is given by:

A = (h/3)[y0 + yn + 4(y1+y3+...+y(n-1)) + 2(y2+y4+...+y(n-2))]

In words this formula can be remembered as:

(h/3)[First + Last + 4(odd numbered) + 2(even numbered]

The First and Last must not appear more than once. Do not include them
in the brackets of even-numbered ordinates.

Example.  Evaluate   INT(0 to 2)[sqrt(x^2+1)dx]

Divide the range into 4 parts each of width 0.5

Calculate the ordinates in a table as follows:

x      sqrt(x^2+1) = y
------------------------
0           1    =  y0
0.5        1.118  =  y1
1         1.4142 =  y2
1.5        1.8027 =  y3
2         2.236  =  y4
-------------------------

A = (.5/3)[1 + 2.236 + 4(1.118 + 1.8027) + 2(1.4142)]

= (1/6)[17.7472]

= 2.95786

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 09/20/97 at 16:57:07
From: keith holm
Subject: Re: Simpson's rule

Thank you very much, Doctor Anthony and the Math Forum; you were a
great help! And I appreciate the fast response time too!  Thanks
again.
```
Associated Topics:
High School Calculus

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