Error of Simpson's, Trapezoid, etc. Rules

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Discussion Overview

The discussion revolves around the error equations associated with numerical integration methods, specifically Simpson's rule and others. Participants explore the origins of these error formulas and the relevance of learning these approximation techniques in the context of functions lacking straightforward antiderivatives.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the source of the error equations for approximation rules like Simpson's rule, expressing curiosity about their derivation.
  • Another participant suggests that learning these rules is important due to the existence of many functions without neat antiderivatives, necessitating numerical integration methods.
  • Some participants mention that various books on numerical integration provide derivations of these formulas, listing specific titles as references.
  • A request is made for someone to share the derivation of the error formulas, indicating a lack of access to comprehensive resources.
  • One participant finds a derivation online but notes that it appears complicated, suggesting that it may be challenging to understand.

Areas of Agreement / Disagreement

Participants generally agree on the importance of numerical integration methods for functions without simple antiderivatives. However, there is no consensus on the specifics of the derivation of error equations, and the complexity of these derivations remains a point of contention.

Contextual Notes

Limitations include the potential complexity of the derivations mentioned and the varying levels of access to educational resources among participants.

iRaid
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I learned this a while ago in my calculus 2 class and I was just thinking about it... Where do these error equations for each of the approximation rules come from? Like for example, where does the error for the Simpson's rule, $$|E_{S}|\le \frac{M(b-a)^{5}}{180n^{4}}$$ come from?

Also, why even bother learning these rules, errors, etc. when you can just compute the actual area more efficiently and effectively with a definite integral?
 
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iRaid said:
I learned this a while ago in my calculus 2 class and I was just thinking about it... Where do these error equations for each of the approximation rules come from? Like for example, where does the error for the Simpson's rule, $$|E_{S}|\le \frac{M(b-a)^{5}}{180n^{4}}$$ come from?

Also, why even bother learning these rules, errors, etc. when you can just compute the actual area more efficiently and effectively with a definite integral?
Because there are vastly many functions that don't have nice, neat antiderivatives, so the only recourse is to do numeric integration using Simpsons' Rule, Gaussian Quadrature, etc.
 
Mark44 said:
Because there are vastly many functions that don't have nice, neat antiderivatives, so the only recourse is to do numeric integration using Simpsons' Rule, Gaussian Quadrature, etc.

I understand that, but where do they get the equations from?
 
Most books on numerical integration show how the formulas are derived. A few that I have on hand are
Applied Numerical Analysis, Second Ed., Curtis E Gerald
Computer Approximations, John F. Hart et al.
Elementary Numerical Analysis: An Algorithmic Approach, S. D. Conte and Carl de Boor
 
Mark44 said:
Most books on numerical integration show how the formulas are derived. A few that I have on hand are
Applied Numerical Analysis, Second Ed., Curtis E Gerald
Computer Approximations, John F. Hart et al.
Elementary Numerical Analysis: An Algorithmic Approach, S. D. Conte and Carl de Boor

Do you mind posting the derivation? I only have one calculus book (Stewart) and it doesn't have it.
 
Try a search online. Someone almost certainly has it posted already.
 
Ok I found a derivation online, it seems very complicated though lol. I see why my book would leave that out.
 

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