What is the difference between trapezoidal rule and simpson rule?


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The trapezoidal rule approximates the integral of the function f(x) on the interval [a, b] as
  (trapezoidal rule)  (b-a)(f(a)+f(b))/2
Simpson's rule uses a second order interpolation to approximate that integral as
  (Simpson's rule)  (b-a)(f(a) + 4f((a+b)/2) + f(b))/6

In regions where the first derivative is not substantially constant, Simpson's rule will give better error performance at the expense of an additional evaluation of the function in each interval.

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