Error of Simpson's, Trapezoid, etc. Rules

[iRaid]

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?

 

[Mark44]

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.

 

[iRaid]

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?

 

[Mark44]

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

 

[iRaid]

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.

 

[Mark44]

Try a search online. Someone almost certainly has it posted already.

 

[iRaid]

Ok I found a derivation online, it seems very complicated though lol. I see why my book would leave that out.