Newton-Cotes Formulae - Maple Help

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Newton-Cotes Formulae

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

ApproximateInt(f(x), x = a..b, method = newtoncotes[N], opts)

ApproximateInt(f(x), a..b, method = newtoncotes[N], opts)

ApproximateInt(Int(f(x), x = a..b), method = newtoncotes[N], opts)

Parameters

f(x)

-

algebraic expression in variable 'x'

x

-

name; specify the independent variable

a, b

-

algebraic expressions; specify the interval

N

-

positive integer

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 = newtoncotes[N], opts) command approximates the integral of f(x) from a to b by using the Nth order Newton-Cotes formula. 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, the Nth order Newton-Cotes formula approximates the integral on each subinterval xi1,xi by integrating the Nth order polynomial which interpolates N1 equally spaced points between the end points of the interval.

  

The Newton-Cotes formulae are generalizations of the simpler polynomial interpolation routines.  The following table gives the correspondence between the other methods and the order.

Equivalent Method

Order

Trapezoid

1

Simpson's Rule

2

Simpson's 3/8 Rule

3

Boole's Rule

4

• 

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.

Examples

∫0.05.0sinxⅆx

0.7163378145

(1)

withStudent[Calculus1]:

ApproximateIntsinx,x=0.0..5.0,method=newtoncotes1

0.7013515555

(2)

ApproximateIntsinx,x=0.0..5.0,method=newtoncotes2

0.7163534765

(3)

ApproximateIntsinx,x=0.0..5.0,method=newtoncotes3

0.7163447696

(4)

ApproximateIntsinx,x=0.0..5.0,method=newtoncotes4

0.7163378087

(5)

ApproximateIntsinx,x=0.0..5.0,method=newtoncotes6

0.7163378145

(6)

ApproximateIntxx2x3,0..5,method=simpson,output=plot

ApproximateInttanx2x,x=1..1,method=simpson,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=simpson,output=animation

See Also

Boole's Rules

plot/options

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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