Simpson’s Rule （辛普森法則）

  In Simpson’s Rule, we will use parabolas to approximate each part of the curve. This proves to be very efficient since it’s generally more accurate than the other numerical methods such as straight lines or trapezoid.

We divide the area into $n$ equal segments of width $\Delta{x}$. The approximate area is given by the following:

Note: In Simpson’s Rule, $n$ must be EVEN.
We can re-write Simpson’s Rule by grouping it as follows:

This gives us an easy way to remember Simpson’s Rule:

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Simpson法則的另一優點在於其自然引出了一種算法，即通過迭代使積分達到所需要的精確度。當積分的上下限相對於展開的中心點對稱時，積分泰勒展開式中含有f(x)的奇數階導數的項都將等於零。利用這一性質，我們可以在相鄰的兩個子區間內對面積作泰勒級數展開。