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I am reading about error bounds with respect to methods of approximating definite integrals such as the trapezoidal rule, simpsons rule, etc. I am trying to derive the error bound for any of the methods of approximation.

For example, for trapezoidal rule:

$\left|E_{T}\right| \leq k \frac{(b-a)^{3}}{12 n^{2}}$

How can I derive this? I started with:

$|E_{T}| = \int_{a}^{b} f(x) d x - T_{n}$

$|E_{T}| = \int_{a}^{b} f(x) d x -\frac{\Delta x}{2}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+2 f\left(x_{2}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]$

I feel the end result can be derived from the Lagrange error bounds. According to which the remainder is given by:

$R=\frac{f^{(n+1)}(c)}{(n+1) !}(x-a)^{n+1}$

How can I manipulate $ \int_{a}^{b} f(x) d x -\frac{\Delta x}{2}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+2 f\left(x_{2}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]$ to use the remainder term formula?

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  • $\begingroup$ The local and global error estimates for the trapezoidal rule are derived in this answer. The notation $h = (b-a)/n$ is used. $\endgroup$ – RRL Nov 8 at 15:32

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