About the error bound of Simpson's rule To find the error bound formula of Simpson's rule with three equally spaced points $x_{0},x_{1},x_{2}$, the error bound of the newton forward polynomial of degree two should be integrated.
As you can see in the attached photo, the integration of the error bound equals zero, how the error can be obtained from the next term in the Newton polynomial??
is this error(corresponding to the Newton polynomial of degree THREE) is appropriate to Simpson's rule with three points?!
Could anyone explain? thanks!
 A: The error term in quadrature formulas is much trickier than is implied in the Wikiversity page quoted above. The first problem is that in the second formula, $\xi=\xi(s)$ is a function of $s$ where $0<\xi(s)<3$ for $s\in(0,2)$ so you would have to look outside the interval of integration to find the error bound. The second problem is that the function $s(s-1)(s-2)(s-3)$ changes sign in the interval $(0,2)$ and this means that the assumption that the error is of the form $$Ch^5f^{(4)}(\xi)$$
is not a given. I showed this in https://math.stackexchange.com/q/1770264 and gave a fix in https://math.stackexchange.com/q/1769300 .  
The summary information of those two posts is that the form for the error in Simpson's rule is correct and can be readily proven to be so by considering an interpolation polynomial that matches $p(x_k)=f(x_k)$ for $x_k\in\{0,1,2\}$ and also $p^{\prime}(x_1)=f^{\prime}(x_1)$ so the polynomial becomes $s(s-1)^2(s-2)$ and doesn't change sign in $(0,2)$ and has 3 zeros in [0,2], one of which is a double zero. Once you know that the form for the error is correct you can start with any trial fourth-degree polynomial and find the constant $C$ above, but proving the correctness of the form is the hard part!
