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

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• Potatochip911
In summary, the Simpson's Rule wikipedia page discusses how the calculation process can be simplified by setting the limits of integration to -1 and 1. This is due to a change of variables known as scaling, which transforms the integral to a simpler form. However, the language used on the page can be confusing and should instead be referred to as a "scaling and shift" or "translation". The quadratic interpolation method, used in Simpson's Rule, can also be applied to other problems in mathematics.
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.

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, Potatochip911 and PeroK
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.)

Potatochip911

## 1. Why is Simpson's rule commonly used for numerical integration?

Simpson's rule is commonly used for numerical integration because it provides a more accurate approximation of the integral compared to other methods, such as the trapezoidal rule or midpoint rule. It also requires fewer function evaluations, making it more efficient for computational purposes.

## 2. What does WLOG mean in the context of deriving Simpson's rule?

WLOG stands for "without loss of generality" and is used in mathematical proofs to show that a particular case can be generalized to a wider range of cases without changing the overall result. In the context of deriving Simpson's rule, it means that we can assume a specific interval, such as -1 to 1, without changing the overall process of deriving the formula.

## 3. How is Simpson's rule derived over a specific interval, such as -1 to 1?

To derive Simpson's rule over a specific interval, we first divide the interval into an even number of subintervals and approximate the integral using a quadratic polynomial for each of these subintervals. We then use these polynomials to create a composite rule that takes into account the curvature of the function over the entire interval, resulting in the Simpson's rule formula.

## 4. Can Simpson's rule be used for any type of function?

Simpson's rule can be used for any function that is continuous over the interval of integration. However, it may not always provide an accurate approximation if the function has sharp changes or is highly oscillatory over the interval. In these cases, other methods may be more suitable.

## 5. Are there any limitations to using Simpson's rule for numerical integration?

One limitation of using Simpson's rule is that it requires an even number of subintervals, which may not always be possible or practical. Additionally, it may not be suitable for highly complex functions that cannot be easily approximated by a quadratic polynomial. In these cases, other numerical integration methods may be more appropriate.

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