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I Why can we WLOG derive Simpson's rule over interval -1 to 1

  1. Nov 30, 2017 #1
    On the Simpson's Rule wikipedia page they mention in their derivation that the calculation can be simplified if one notices that there is no loss in generality in setting ##a=-1## and ##b=1## for the integral ##\int_{a}^{b}P(x)\cdot dx## as a result of scaling.

    I'm not entirely sure what they're referring to by scaling but if I had to guess it would be applying Simpson's rule to ##n## sub-intervals. I'm also thrown off by the fact that any odd function will just be 0 over this interval whereas over an asymmetric interval it would in general not be 0 yet somehow this is all valid.
     
  2. jcsd
  3. Dec 1, 2017 #2
    By scaling, they mean a change of variables of the form
    $$x=(b-a)x'/2+(a+b)/2,$$
    where ##x'\in[-1,1]##.
    By this change of variables an integral on [a,b] can be transformed into one on [-1,1].
     
  4. Dec 1, 2017 #3

    StoneTemplePython

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    Gold Member

    The language in wikipedia is a bit sloppy. It really should be called a "scaling and shift / translation". If we were in a vector space, the translation from the origin would put this in an affine space. Sometimes that matters a lot, sometimes not.

    Note: the quadratic interpolation method is quite powerful and shows up elsewhere. (E.g. in combination with Rolle's theorem a simple quadratic approximation can prove many useful results related to convexity.)
     
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