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

[Potatochip911]

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.

 

[eys_physics]

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].

 

[StoneTemplePython]

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.)