# Error of Simpson's, Trapezoid, etc. Rules

1. Nov 11, 2013

### 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?

2. Nov 12, 2013

### Staff: Mentor

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.

3. Nov 12, 2013

### iRaid

I understand that, but where do they get the equations from?

4. Nov 12, 2013

### Staff: Mentor

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

5. Nov 12, 2013

### iRaid

Do you mind posting the derivation? I only have one calculus book (Stewart) and it doesn't have it.

6. Nov 13, 2013

### Staff: Mentor

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

7. Nov 13, 2013

### iRaid

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