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/