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Newton-Cotes Formulae
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Calling Sequence
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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)
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Parameters
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f(x)
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algebraic expression in variable 'x'
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x
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name; specify the independent variable
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a, b
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algebraic expressions; specify the interval
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N
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positive integer
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opts
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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
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Description
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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.
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If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.
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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.
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Equivalent Method
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Order
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Trapezoid
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1
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Simpson's Rule
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2
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Simpson's 3/8 Rule
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3
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Boole's Rule
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4
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By default, the interval is divided into  equal-sized subintervals.
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Examples
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>
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| (1) |
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![with(Student[Calculus1])](/support/helpjp/helpview.aspx?si=4357/file04932/math239.png)
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![ApproximateInt(sin(x), x = 0. .. 5.0, method = newtoncotes[1])](/support/helpjp/helpview.aspx?si=4357/file04932/math243.png)
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| (2) |
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![ApproximateInt(sin(x), x = 0. .. 5.0, method = newtoncotes[2])](/support/helpjp/helpview.aspx?si=4357/file04932/math250.png)
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| (3) |
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![ApproximateInt(sin(x), x = 0. .. 5.0, method = newtoncotes[3])](/support/helpjp/helpview.aspx?si=4357/file04932/math257.png)
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| (4) |
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![ApproximateInt(sin(x), x = 0. .. 5.0, method = newtoncotes[4])](/support/helpjp/helpview.aspx?si=4357/file04932/math264.png)
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| (5) |
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![ApproximateInt(sin(x), x = 0. .. 5.0, method = newtoncotes[6])](/support/helpjp/helpview.aspx?si=4357/file04932/math271.png)
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| (6) |
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See Also
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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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