# Prove Simpson's rule (including error) using the integral remainder

I have to prove Simpson's rule including the error with the help of the integral remainder. However, I have practically no idea how to start.

Let $f: [a,b] \rightarrow \mathbb{R}$ be continuously differentiable four times. Let $n \in 2\mathbb{N}$ and $h = (b-a)/n$. Let $y_k = f(a+kh)$ for $0 \leq k \leq n$. Show:

$$\left|\int_a^b f(x) \mathrm dx - \frac{h}{3} [(y_0 + y_n) + 4(y_1+y_3+...+y_{n-1})+2(y_2+y_4+...+y_{n-2})]\right|$$ $$\leq \frac{1}{180} \max_{a \leq x \leq b} |f^{(4)}(x)|(b-a)h^4$$

I thought I'd do a Taylor expansion for $\int_a^b f(x) \mathrm dx$ and then apply the integral remainder, but then all terms vanish and I get $0=0$ which is not very helpful...

I only need a good advice how to start! Thanks in advance!

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Have you tried proving it for the case of one panel? – J. M. Dec 14 '10 at 16:09
No, but the exercise is as written and we never had the "normal case", so I guess there should be a way to get to the above inequality directly, isn't there? – Huy Dec 14 '10 at 16:18
Supposedly, your first order of business is to prove that Simpson's rule exactly integrates polynomials of degree 3 and lower... – J. M. Dec 14 '10 at 16:49