Deriving Simpson's Rule

Date: 06/07/98 at 16:11:29
From: Carol
Subject: Calculus - Simpson's Rule

I understand that using Simpson's rule is easier than using the 
Trapezoidal formula in order to find the area under a curve. I wanted 
to know how you derive the rule. I'm taking calculus AB and although 
my AB textbook mentions Simpson and his formula, it doesn't go into 
detail about how the rule was derived.

Date: 06/07/98 at 18:11:06
From: Doctor Pat
Subject: Re: Calculus - Simpson's Rule

Carol,

I don't know that Simpson's rule is "easier," but it is a quadratic 
rather than a linear approach, and therefore should be a little more 
accurate. Anyway, here is the derivation for the "rule."

The Simpson method makes a parabola through three consecutive points, 
so we begin with the general equation of a parabola:

   y = Ax^2 + Bx + C
  
and the area from x = -h to x = h is given by: 
 
  Area = INT(-h to h) [Ax^2 + Bx + C] dx  

which is:

    Ax^3       Bx^2        
   ------  +  ------  +  Cx   from -h to h
     3          2

This simplifies to (2Ah^3)/3 + 2Ch and by some really tricky 
factoring becomes:

   (h/3)(2Ah^2 +6C)  
   
Now using the fact that the curve goes through (-h,y0), (0,y1), and 
(h,y2), we can substitute and get:

   y0 = Ah^2 - Bh + C         
   y1 = C    
   y2 = Ah^2 + Bh + C

giving:

    C = y1 

and:

   Ah^2 - Bh = y0 - y1   
   Ah^2 + Bh = y2 - y1
   ----------------------
   2 Ah^2 = y0 + y2 - 2y1

Now we plug these into the Area integral above and get:

   Area = (h/3)[2Ah^2 + 6C] = (h/3)[y0 + y2 - 2y1 + 6y1]
        = (h/3)[y0 + 4y1 + y2]   

Simpson's rule just applies this formula to successive pieces of the 
curve like this:

     (h/3)[ y0 + 4y1 + y2]  
   + (h/3)[ y2 + 4y3 + y4] 
   + (h/3)[ y4 + 4y5 + y6] 

Now when we factor h/3 and combine the terms under addition we get:

   (h/3)[y0 + 4y1 + 2y2 + 4y3 + 2y4 + 4y5 + y6]

and the deed is done. 

Hope that was clear. Maybe a good book in the library would have some 
pictures to help. Thanks for writing.    

-Doctor Pat,  The Math Forum
 http://mathforum.org/dr.math/