Associated Topics || Dr. Math Home || Search Dr. Math

Simpson's Rule/Trapezoid Rule

```
Date: 02/01/98 at 01:08:08
From: Allen Riddell
Subject: Simpson's Rule/Trapezoidal Rule (Formula) for
approximating integrals.

I understand why the trapezoidal rule results in a better
approximation of an integral. What I am uncertain about is how
Simpson's rule is derived and why it is a better approximation of the
integral than the midpoint (rule?) or trapezodial method. I'm finding
it difficult to cope with my teacher's assertion that "it just is."

I understand that Simpson's rule gets an approximation by using a
parabola. What I'm completely confused on is why a parabola is on
balance a better approximation than the trapazoid. Also - I don't see
any x^2 or anything that suggests a curve in the actual equation that
represents Simpson's rule. Jacob Krich is Swarthmore god!
```

```
Date: 02/01/98 at 08:45:12
From: Doctor Jerry
Subject: Re: Simpson's Rule/Trapezoidal Rule (Formula) for
approximating integrals.

Hi Allen,

Perhaps an example will help.

Let f(x)=e^x and consider int(0,1,e^x*dx). Divide the interval [0,1]
into two parts with the subdivision {0,0.5,1}. The trapezoid rule can
be seen as fitting a line to the curve between (0,0) and (0.5,e^0.5)
and a second line between (0.5,e^0.5) and (1,e) and then, instead of
integrating e^x over these two intervals, we integrate the equations
of the two lines. Simpson's Rule can be seen as fitting a parabola to
the three points (0,1), (0.5,e^0.5), and (1,e) and then, instead of
integating e^x over these two intervals, we integrate the equation of
the parabola.

To my eye, the parabola does a better job of "snugging up" to the
curve than the two lines. Here are some details:

Integrating the two lines gives 1/4+sqrt(e)/2+e/4, which is 1.75393...
This is the Trapezoid Rule.

Integrating the parabola gives 1/6+2sqrt(e)/3+e/6, which is 1.71886...
This is Simpson's Rule.

The exact answer is e-1, which is 1.71828...

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search