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Boole's Rule

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

ApproximateInt(f(x), x = a..b, method = boole, opts)

ApproximateInt(f(x), a..b, method = boole, opts)

ApproximateInt(Int(f(x), x = a..b), method = boole, opts)

Parameters

f(x)

-

algebraic expression in variable 'x'

x

-

name; specify the independent variable

a, b

-

algebraic expressions; specify the interval

opts

-

equation(s) of the form option=value where option is one of boxoptions, functionoptions, iterations, method, outline, output, partition, pointoptions, refinement, showarea, showfunction, showpoints, subpartition, view, or Student plot options; specify output options

Description

• 

The ApproximateInt(f(x), x = a..b, method = boole, opts) command approximates the integral of f(x) from a to b by using Boole's rule. The first two arguments (function expression and range) can be replaced by a definite integral.

• 

If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.

• 

Given a partition P=a=x0,x1,...,xN=b of the interval a,b, Boole's rule approximates the integral on each subinterval xi1,xi by integrating the quartic function that interpolates five equally spaced points in that subinterval.

• 

In the case that the widths of the subintervals are equal, the approximation can be written as

  

 

18bafx+3f23x0+13x1+3f13x0+23x1+2fx1+3f23x1+13x2+3f13x1+23xw+2fx2+...+3f13xN1+23xN+fxNN

  

Traditionally, Boole's rule is written as: given N, where N is a positive multiple of 3, and given equally spaced points a=x0,x1,x2,...,xN=b, an approximation to the integral ∫abfxⅆx is

  

 

38bafx0+3fx1+3fx2+2fx3+3fx4+3fx5+2fx6+3fx7+...+3fxN1+fxNN

• 

By default, the interval is divided into 10 equal-sized subintervals.

• 

For the options opts, see the ApproximateInt help page.

• 

This rule can be applied interactively, through the ApproximateInt Tutor.

• 

This rule is also sometimes known as Bode's Rule due to a misattribution in the literature.  The ApproximateInt command will accept either method=boole or method=bode.

Examples

polynomialCurveFitting[PolynomialInterpolation]x0,3x0+x14,x0+x22,x0+3x14,x1,f0,f14,f12,f34,f1,z:

integrated∫x0x1polynomialⅆz:

factorintegrated

190x0x188f1x1x23624f34x1x23+72f12x02x12656f14x1x2372f0x1x23+72f1x02x1x2114f1x0x12x2+42f1x0x1x22+336f34x02x1x2768f34x0x12x2+432f34x0x1x22+432f14x02x1x2768f14x0x12x2+336f14x0x1x22+60f0x02x1x2150f0x0x12x2+90f0x0x1x2226f1x03x12f1x03x2+3f1x02x1233f1x02x22+36f1x0x13+8f1x0x2336f1x13x2+111f1x12x22144f34x03x1+16f34x03x2+48f34x02x12192f34x02x22+224f34x0x1316f34x0x23224f34x13x2+720f34x12x2248f12x03x148f12x0x13112f14x03x116f14x03x248f14x02x12192f14x02x22+288f14x0x13+16f14x0x23288f14x13x2+816f14x12x2230f0x03x1+2f0x03x2+15f0x02x1233f0x02x22+40f0x0x138f0x0x23+8f0x13x2+63f0x12x22+7f1x04+20f1x24+32f34x04+160f34x24+12f12x04+12f12x14+32f14x04+160f14x24+7f0x0412f0x14+20f0x24x03x1+2x2x02x2+x1x0x22x1+x0+x2

(1)

withStudent[Calculus1]:

ApproximateIntsinx,x=0..5,method=boole

115sin194+845sin398+7180sin5+790sin4+845sin338+115sin174+845sin358+790sin92+845sin378+115sin134+845sin278+790sin72+845sin298+115sin154+845sin318+845sin198+790sin52+845sin218+115sin114+845sin238+790sin3+845sin258+845sin138+115sin74+845sin158+790sin2+845sin178+115sin94+845sin78+845sin98+115sin54+845sin118+790sin32+115sin14+845sin38+790sin12+845sin58+115sin34+845sin18+790sin1

(2)

ApproximateIntxx2x3,x=0..5,method=boole,output=plot

ApproximateInttanx2x,1..1,method=boole,output=plot,partition=50

To play the following animation in this help page, right-click (Control-click, on Macintosh) the plot to display the context menu.  Select Animation > Play.

ApproximateIntlnx,x=1..100,method=boole,output=animation

See Also

ApproximateInt

int

Newton-Cotes Rules

Simpson's 3/8 Rule

Simpson's Rule

Student

Student plot options

Student[Calculus1]

Student[Calculus1][ApproximateInt]

Student[Calculus1][ApproximateIntTutor]

Student[Calculus1][RiemannSum]

Student[Calculus1][VisualizationOverview]

Trapezoidal Rule

 


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