linguistic:paraphrasing, linguistic:voice_change
Conservative (a priori) error estimation

Let f {\displaystyle f} have a bounded first derivative over, {\displaystyle,} i.e. f ∈ C 1 (). {\displaystyle f\in C^{1}().} The mean value theorem for f, {\displaystyle f,} where x ∈ [ a, b), {\displaystyle x\in [a,b),} gives

for some ξ x ∈ (a, x ] {\displaystyle \xi _{x}\in (a,x]} depending on x {\displaystyle x}.

If we integrate in x {\displaystyle x} from a {\displaystyle a} to b {\displaystyle b} on both sides and take the absolute values, we obtain

We can further approximate the integral on the right-hand side by bringing the absolute value into the integrand, and replacing the term in f ′ {\displaystyle f'} by an upper bound

where the supremum was used to approximate.

Hence, if we approximate the integral ∫ a b f (x) d x {\textstyle \int _{a}^{b}f(x)\,dx} by the quadrature rule (b − a) f (a) {\displaystyle (b-a)f(a)} our error is no greater than the right hand side of 1. We can convert this into an error analysis for the Riemann sum, giving an upper bound of

for the error term of that particular approximation. (Note that this is precisely the error we calculated for the example f (x) = x {\displaystyle f(x)=x}.) Using more derivatives, and by tweaking the quadrature, we can do a similar error analysis using a Taylor series (using a partial sum with remainder term) for f. This error analysis gives a strict upper bound on the error, if the derivatives of f are available.

This integration method can be combined with interval arithmetic to produce computer proofs and verified calculations.