6. Simpson's Rule

by M. Bourne

The Trapezoidal Rule is an improvement over using rectangles because we have much less "missing" from our calculations. We used straight lines to model the curve in trapezoidal Rule.

We seek an even better approximation. In Simpson's Rule, we use parabolas to approximate each part of the curve. This proves to be very efficient. (See more about Parabolas.)

Area under a curve using Simpson's Rule

We can show (by integrating the area under each parabola and adding these areas) that the approximate area is given the following. This is called Simpson's Rule:

`"Area"=int_a^bf(x)dx`

`~~(Deltax)/3(y_0+4y_1+2y_2+4y_3+2y_4...+4y_(n-1)+y_n)`

Note: In Simpson's Rule, n must be EVEN.

Example

Approximate `int_2^3(dx)/(x+1`using Simpson's Rule with `n=4`.

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