The numerical integration technique known as "Simpson's 3/8 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 3/8 Rule)  Consider over , where , , and .  Simpson's 3/8 rule is   

    .   

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

    .  

The remainder term for Simpson's 3/8 rule is  ,  where lies somewhere between , and have the equality  

    .

Proof  Simpson's 3/8 Rule  Simpson's 3/8 Rule

Composite Simpson's 3/8 Rule

    Our next method of finding the area under a curve is by approximating that curve with a series of cubic segments that lie above the intervals  .  When several cubics are used, we call it the composite Simpson's 3/8 rule.  

Theorem (Composite Simpson's 3/8 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 3/8 rule for subintervals  is  

    .  

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

    .  

Proof  Simpson's 3/8 Rule  Simpson's 3/8 Rule

Remainder term for the Composite Simpson's 3/8 Rule

Corollary  (Simpson's 3/8 Rule:  Remainder term)   Suppose that is subdivided into subintervals    of width  .  The composite Simpson's 3/8 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's 3/8 Rule.  To approximate the integral  

    ,  


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

Animations (Simpson's 3/8 Rule  Simpson's 3/8 Rule).  Internet hyperlinks to animations.

Computer Programs  Simpson's 3/8 Rule  Simpson's 3/8 Rule  

Mathematica Subroutine (Simpson's 3/8 Rule). Object oriented programming.

Example 1.  Numerically approximate the integral    by using Simpson's 3/8 rule with  m = 1, 2, 4.
Solution 1.

Example 2.  Numerically approximate the integral    by using Simpson's 3/8 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' 3/8 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 3/8 rule with  m = 1, 2, 4.
Solution 6.

Example 7.  Numerically approximate the integral    by using Simpson's 3/8 rule with  m = 10, 20, 40, 80,  and 160.
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's 3/8 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 3/8 Rule.

Example 11.   Let    over  .  Use Simpson's 3/8 rule to approximate the value of the integral.
Solution 11.

Animations (Simpson's 3/8 Rule  Simpson's 3/8 Rule).  Internet hyperlinks to animations.

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 3/8 Rule for Numerical Integration

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(c) John H. Mathews 2004