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

    .   

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

    .  

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

    .

Proof  Boole's Rule  Boole's Rule  

Composite Boole's Rule

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

Theorem (Composite Boole'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 Boole's rule for subintervals  is  

    .  

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

    .  

Proof  Boole's Rule  Boole's Rule  

Remainder term for the Composite Boole's Rule

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

    ,  

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

Animations (Boole's Rule  Boole's Rule).  Internet hyperlinks to animations.

Computer Programs  Boole's Rule  Boole's Rule  

Mathematica Subroutine (Boole's Rule). Object oriented programming.

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

Example 2.  Numerically approximate the integral    by using Boole's rule with  m = 1, 2, 4, 8,  and 16.
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 Boole's 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 Boole's rule with  m = 1, and 2.
Solution 6.

Example 7.  Numerically approximate the integral    by using Boole's rule with  m = 1, 2, 4, 8, 16, and 32.
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 Boole's 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 Boole's Rule.

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

Animations (Boole's Rule  Boole's Rule).  Internet hyperlinks to animations.

Download this Mathematica Notebook Boole's Rule for Numerical Integration

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(c) John H. Mathews 2004