# The Trapezoid rule and Simpson's rule: numerical integration

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Added by astrobookcat1 on May 06, 2002 at 20:23:24:

To find values for difficult integrals, either the Trapezoid rule and Simpson's rule can be used. The trapezoid rule approximates area under curves using (logically enough) trapezoids, and Simpson's rule approximates using parabolas. Simpson's rule is generally more accurate and is exact if the polynomial being integrated is of degree three or less.

?x= (b-a)/n, where a and b are the limits of the integral and n is the number of subintervals used

Trapezoid rule: A= (?x/2)[f(x0) +2f(x1) +2f(x2) . . . +2f(xn-1) +f(xn)]
Trapezoid error: E<= [((b-a)^3)M2]/(12n^2) where M2 is the maximum of the second derivitive of f on the interval [a,b] and n is the number of subintervals

Simpson's rule: A= (?x/3)[f(x0) + 4f(x1) +2f(x2) +4f(x4). . . +2f(xn-2) +4f(xn-1) +f(xn)]
Simpson's error: E<=[((b-a)^5)M4]/(180n^4) where M4 is the maximum of the fourth derivative of f on the interval [a,b] and n is the number of subintervals