Boole's Rule

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  

    .

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  

    .  

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  .  

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