# Derivation of error bounds for methods of approximating definite integrals

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?

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