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for

Boole's Rule for Numerical Integration

   

Theorem  (Boole's Rule)  Consider [Graphics:Images/BooleRuleMod_gr_1.gif] over [Graphics:Images/BooleRuleMod_gr_2.gif], where [Graphics:Images/BooleRuleMod_gr_3.gif], [Graphics:Images/BooleRuleMod_gr_4.gif], [Graphics:Images/BooleRuleMod_gr_5.gif], and [Graphics:Images/BooleRuleMod_gr_6.gif].  Boole's rule is   

    
[Graphics:Images/BooleRuleMod_gr_7.gif][Graphics:Images/BooleRuleMod_gr_8.gif].   

This is an numerical approximation to the integral of
[Graphics:Images/BooleRuleMod_gr_9.gif] over [Graphics:Images/BooleRuleMod_gr_10.gif] and we have the expression  

    [Graphics:Images/BooleRuleMod_gr_11.gif].  

The remainder term for Boole's rule is  [Graphics:Images/BooleRuleMod_gr_12.gif],  where [Graphics:Images/BooleRuleMod_gr_13.gif] lies somewhere between [Graphics:Images/BooleRuleMod_gr_14.gif], and have the equality  

    [Graphics:Images/BooleRuleMod_gr_15.gif].

Proof  Boole's Rule  Boole's Rule  

 

Composite Boole's Rule

    Our next method of finding the area under a curve [Graphics:Images/BooleRuleMod_gr_16.gif] is by approximating that curve with a series of quartic segments that lie above the intervals  [Graphics:Images/BooleRuleMod_gr_17.gif].  When several parabolas are used, we call it the composite Boole's rule.  

 

Theorem (Composite Boole's Rule)  Consider [Graphics:Images/BooleRuleMod_gr_18.gif] over [Graphics:Images/BooleRuleMod_gr_19.gif].  Suppose that the interval [Graphics:Images/BooleRuleMod_gr_20.gif] is subdivided into [Graphics:Images/BooleRuleMod_gr_21.gif] subintervals  [Graphics:Images/BooleRuleMod_gr_22.gif]  of equal width  [Graphics:Images/BooleRuleMod_gr_23.gif]  by using the equally spaced sample points  [Graphics:Images/BooleRuleMod_gr_24.gif]  for  [Graphics:Images/BooleRuleMod_gr_25.gif].   The composite Boole's rule for [Graphics:Images/BooleRuleMod_gr_26.gif] subintervals  is  

    
[Graphics:Images/BooleRuleMod_gr_27.gif][Graphics:Images/BooleRuleMod_gr_28.gif][Graphics:Images/BooleRuleMod_gr_29.gif].  

This is an numerical approximation to the integral of
[Graphics:Images/BooleRuleMod_gr_30.gif] over [Graphics:Images/BooleRuleMod_gr_31.gif] and we write  

    [Graphics:Images/BooleRuleMod_gr_32.gif].  

Proof  Boole's Rule  Boole's Rule  

 

Remainder term for the Composite Boole's Rule

Corollary  (Boole's Rule:  Remainder term)   Suppose that [Graphics:Images/BooleRuleMod_gr_33.gif] is subdivided into [Graphics:Images/BooleRuleMod_gr_34.gif] subintervals  [Graphics:Images/BooleRuleMod_gr_35.gif]  of width  [Graphics:Images/BooleRuleMod_gr_36.gif].  The composite Boole's rule  

    
[Graphics:Images/BooleRuleMod_gr_37.gif][Graphics:Images/BooleRuleMod_gr_38.gif][Graphics:Images/BooleRuleMod_gr_39.gif].  

is an numerical approximation to the integral, and  

    
[Graphics:Images/BooleRuleMod_gr_40.gif].  

Furthermore, if [Graphics:Images/BooleRuleMod_gr_41.gif],  then there exists a value [Graphics:Images/BooleRuleMod_gr_42.gif] with  [Graphics:Images/BooleRuleMod_gr_43.gif]  so that the error term  [Graphics:Images/BooleRuleMod_gr_44.gif]  has the form

    [Graphics:Images/BooleRuleMod_gr_45.gif].  

This is expressed using the "big [Graphics:Images/BooleRuleMod_gr_46.gif]" notation  [Graphics:Images/BooleRuleMod_gr_47.gif].  

 

Remark.  When the step size is reduced by a factor of [Graphics:Images/BooleRuleMod_gr_48.gif] the remainder term  [Graphics:Images/BooleRuleMod_gr_49.gif] should be reduced by approximately [Graphics:Images/BooleRuleMod_gr_50.gif].  

 

Algorithm Composite Boole's Rule.  To approximate the integral  

    [Graphics:Images/BooleRuleMod_gr_51.gif],  

by sampling  [Graphics:Images/BooleRuleMod_gr_52.gif]  at the  [Graphics:Images/BooleRuleMod_gr_53.gif]  equally spaced sample points  [Graphics:Images/BooleRuleMod_gr_54.gif] for  [Graphics:Images/BooleRuleMod_gr_55.gif],  where  [Graphics:Images/BooleRuleMod_gr_56.gif].  Notice that  [Graphics:Images/BooleRuleMod_gr_57.gif]  and  [Graphics:Images/BooleRuleMod_gr_58.gif].  

 

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.

[Graphics:Images/BooleRuleMod_gr_59.gif]

Example 1.  Numerically approximate the integral  [Graphics:Images/BooleRuleMod_gr_60.gif]  by using Boole's rule with  m = 1, and 2.
Solution 1.

 

Example 2.  Numerically approximate the integral  [Graphics:Images/BooleRuleMod_gr_71.gif]  by using Boole's rule with  m = 1, 2, 4, 8,  and 16.
Solution 2.

 

Example 3.  Find the analytic value of the integral  [Graphics:Images/BooleRuleMod_gr_88.gif]  (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 [Graphics:Images/BooleRuleMod_gr_108.gif] the error term  [Graphics:Images/BooleRuleMod_gr_109.gif] should be reduced by approximately [Graphics:Images/BooleRuleMod_gr_110.gif].  Explore this phenomenon.
Solution 5.

 

Example 6.  Numerically approximate the integral [Graphics:Images/BooleRuleMod_gr_119.gif] by using Boole's rule with  m = 1, and 2.
Solution 6.

 

Example 7.  Numerically approximate the integral  [Graphics:Images/BooleRuleMod_gr_129.gif]  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  [Graphics:Images/BooleRuleMod_gr_149.gif]  (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 [Graphics:Images/BooleRuleMod_gr_168.gif] the error term  [Graphics:Images/BooleRuleMod_gr_169.gif] should be reduced by approximately [Graphics:Images/BooleRuleMod_gr_170.gif].  Explore this phenomenon.
Solution 10.

 

Various Scenarios and Animations for Boole's Rule.

Example 11.   Let  [Graphics:Images/BooleRuleMod_gr_181.gif]  over  [Graphics:Images/BooleRuleMod_gr_182.gif].  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