Numerical Integration with Simpson's Rule

    Instead of rectangles and trapezoids Simpson's rule uses parabolic arcs. The formula obtained for Simpson's rule requires using uniform width subintervals and an even number of them.

    Consider on and say we want to find the area. We could use rectangles as in a Riemann sum; however many graphs have curvature as so maybe using strips with parabolic arc will yield better estimates with less work.

That is instead if rectangles or trapezoids:



let's use "strips" with parabolic arcs



If is a partition of with and and if we pass a parabolic arc through the points, three at a time say, the points with -coordinates then those with and so on. It can be shown that the area of the region under the parabolic curve on the interval has area given by



where Thus, to estimate we can add up the strips as follows:

Definition (Simpson's Rule) The Simpson rule estimates the definite integral of over using the formula,  



where and is an even integer.

Example (Simpson's Rule) Consider on and let   be a partition of the interval (so ).  





which is a much better estimation than using rectangles with the same partition.

Definition (Error in the Simpson Rule) If has a continuous fourth derivative on then the error (n even) in approximating   by the Simpson rule satisfies:



where is the maximum value of on

Definition (Error in the Simpson Rule) Consider on and let   be a partition of the interval . Find the error in the approximation



    Solution. We compute ,  
    




and

.

We plot the fourth derivative obtaining:
    


and so there is a maxmium at We obtain Therefore, the error is less than

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