MATHS Simpson's Rule

冇架﹐積出來的﹐以下是最簡明的証明
We want to derive Simpson's rule for


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We will use Newton and Cotes formulas for
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.
The corresponding interpolating polynomial is


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and thus


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Since integration is linear, we are concerned only with integrating each term in the sum. Now, taking
圖片參考：http://images.planetmath.org:8080/cache/objects/6509/l2h/img13.png
, we can rewrite the quotients on the last integral as


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and if we calculate the integrals on the last expression we get


圖片參考：http://images.planetmath.org:8080/cache/objects/6509/l2h/img15.png

which is Simpson's rule:


圖片參考：http://images.planetmath.org:8080/cache/objects/6509/l2h/img16.png

我都唔係好明唷
難得咁快有人應
可唔可以教下我

中間的strip係double count咗的，所以會有個4出咗嚟。