The Simpson's rule approximation Sn (n an even positive integer) is given for an arbitrary function f by
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Sn | |
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The error bound for Simpson's rule is given by
where K is any number such thatf(4)(x)
≤K for all x∈
a, b
. In particular, Simpson's rule is exact for integrals of polynomials of degree at most 3 (because the fourth derivative of such a polynomial is identically zero).
To approximate
x sin x dx using Simpson's rule
Calculus + Approximate Integral + Simpson
x sin x dx Approximate integral (Simpson's rule) is
π
4
2i - 1
πsin
2i - 1
π +2
iπsin
iπ
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Evaluate Numerically
π
4
2i - 1
πsin
2i - 1
π +2
iπsin
iπ
= 3.1418
Plot 2D + Rectangular
12e-x2 -48x2 +16x4e-x2
dtbpF3in2.0003in0pt