Simpson's Rule for Numerical Integration

    The numerical integration technique known as "Simpson's Rule" is credited to the mathematician Thomas Simpson (1710-1761) of  Leicestershire, England. His also worked in the areas of numerical interpolation and probability theory.

Theorem  (Simpson's Rule)  Consider over , where , and .  Simpson's rule is   

    .   

This is an numerical approximation to the integral of over and we have the expression  

    .  

The remainder term for Simpson's rule is  ,  where lies somewhere between , and have the equality  

    .

Composite Simpson Rule

    Our next method of finding the area under a curve is by approximating that curve with a series of parabolic segments that lie above the intervals  .  When several parabolas are used, we call it the composite Simpson rule.  

Theorem (Composite Simpson's Rule)  Consider over .  Suppose that the interval is subdivided into subintervals    of equal width    by using the equally spaced sample points    for  .   The composite Simpson's rule for subintervals  is  

    .  

This is an numerical approximation to the integral of over and we write  

    .  

Remainder term for the Composite Simpson Rule

Corollary  (Simpson's Rule:  Remainder term)   Suppose that is subdivided into subintervals    of width  .  The composite Simpson's rule  

    .  

is an numerical approximation to the integral, and  

    .  

Furthermore, if ,  then there exists a value with    so that the error term    has the form

    .  

This is expressed using the "big " notation  .  

Remark.  When the step size is reduced by a factor of the remainder term   should be reduced by approximately .  

Algorithm Composite Simpson Rule.  To approximate the integral  

    ,  

by sampling    at the    equally spaced sample points   for  ,  where  .  Notice that    and  .  

Mathematica Subroutine (Simpson Rule). Traditional programming.

Mathematica Subroutine (Simpson Rule). Object oriented programming.

Example  Numerically approximate the integral    by using Simpson's rule with  m = 1, 2, 4, and 8.