what s the most efficient method of approximating definite integral function?trapezoidal rule or simpson rule?why?

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embizze | High School Teacher | (Level 1) Educator Emeritus

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In general Simpson's rule is more efficient, if by efficient you mean having a smaller error for the same number of intervals.

The error for the trapezoidal rule is:

`E<=((b-a)^3)/(12n^2)["max"|f''(x)|],a<=x<=b`

So the error has an upper bound dependent on the extreme values of the second derivative. This error can be made as small as wanted by increasing n (the number of intervals).

The error for Simpson's rule is:

`E<=((b-a)^5)/(180n^4)["max"|f^((4))(x)|],a<=x<=b`

Here the upper bound depends on the extreme value of the fourth derivative. Also notice that as n increases, the error reduces by `n^4` compared to `n^2` for the trapezoidal rule.

Simpson's Rule uses a quadratic to model the function on the interval, while the trapezoidal rule uses a linear function. So the more "curved" the function is, the better Simpson's rule will be compared to the trapezoidal rule.

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