Simpson's 3/8 Rule
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Calling Sequence
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ApproximateInt(f(x), x = a..b, method = simpson[3/8], opts)
ApproximateInt(f(x), a..b, method = simpson[3/8], opts)
ApproximateInt(Int(f(x), x = a..b), method = simpson[3/8], 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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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 = simpson[3/8], opts) command approximates the integral of f(x) from a to b by using Simpson's 3/8 rule. This rule is also known as Newton's 3/8 rule. 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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In the case that the widths of the subintervals are equal, the approximation can be written as
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![(1/8)*(b-a)*(f(x[0])+3*f((2/3)*x[0]+(1/3)*x[1])+3*f((1/3)*x[0]+(2/3)*x[1])+2*f(x[1])+3*f((2/3)*x[1]+(1/3)*x[2])+3*f((1/3)*x[1]+(2/3)*x[2])+2*f(x[2])+`...`+f((1/3)*x[N-1]+(2/3)*x[N])+f(x[N]))/N](/support/helpjp/helpview.aspx?si=4359/file04947/math170.png)
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Traditionally, Simpson's 3/8 rule is written as: given N, where N is a positive multiple of 3, and given equally spaced points , an approximation to the integral is
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![(3/8)*(b-a)*(f(x[0])+3*f(x[1])+3*f(x[2])+2*f(x[3])+3*f(x[4])+3*f(x[5])+2*f(x[6])+3*f(x[7])+`...`+3*f(x[N-1])+f(x[N]))/N](/support/helpjp/helpview.aspx?si=4359/file04947/math185.png)
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By default, the interval is divided into equal-sized subintervals.
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Examples
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See Also
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Boole's Rules, Newton-Cotes Rules, plot/options, 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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