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  

    .

Proof  Simpson's Rule for Numerical Integration  Simpson's Rule for Numerical Integration  

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  

    .  

Proof  Simpson's Rule for Numerical Integration  Simpson's Rule for Numerical Integration  

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  .  

Animations (Simpson's Rule  Simpson's Rule).  

Computer Programs  Simpson's Rule for Numerical Integration  Simpson's Rule for Numerical Integration  

Mathematica Subroutine (Simpson Rule). Traditional programming.

Mathematica Subroutine (Simpson Rule). Object oriented programming.

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

Example 2.  Numerically approximate the integral    by using Simpson's rule with  m = 10, 20, 40, 80,  and 160.
Solution 2.

Example 3.  Find the analytic value of the integral    (i.e. find the "true value").   
Solution 3.

Example 4.  Use the "true value" in example 3 and find the error for the Simpson rule approximations in example 2.  
Solution 4.

Example 5.  When the step size is reduced by a factor of the error term   should be reduced by approximately .  Explore this phenomenon.
Solution 5.

Example 6.  Numerically approximate the integral by using Simpson's rule with  m = 1, 2, 4, and 8.
Solution 6.

Example 7.  Numerically approximate the integral    by using Simpson's rule with  m = 10, 20, 40, 80,  and 160  subintervals.
Solution 7.

Example 8.  Find the analytic value of the integral    (i.e. find the "true value").   
Solution 8.

Example 9.  Use the "true value" in example 8 and find the error for the Simpson rule approximations in example 7.  
Solution 9.

Example 10.  When the step size is reduced by a factor of the error term   should be reduced by approximately .  Explore this phenomenon.
Solution 10.

Various Scenarios and Animations for Simpson's Rule.

Example 11.   Let    over  .  Use Simpson's rule to approximate the value of the integral.
Solution 11.

Animations (Simpson's Rule  Simpson's Rule).  

Old Lab Project (Simpson's Rule  Simpson's Rule).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

Simpson's Rule for Numerical Integration  Simpson's Rule for Numerical Integration  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Simpson's Rule for Numerical Integration

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(c) John H. Mathews 2004