Editing Numerical integration (section)

=== Conservative (a priori) error estimation ===
Let <math>f</math> have a bounded first derivative over <math>[a,b],</math> i.e. <math>f \in C^1([a,b]).</math> The [[mean value theorem]] for <math> f,</math> where <math>x \in [a,b),</math> gives
<math display="block"> (x - a) f'(\xi_x) = f(x) - f(a), </math>
for some <math> \xi_x \in (a,x] </math> depending on <math> x </math>.

If we integrate in <math> x </math> from <math> a </math> to <math> b </math> on both sides and take the absolute values, we obtain
<math display="block">
    \left| \int_a^b f(x)\, dx - (b - a) f(a) \right|
  = \left| \int_a^b (x - a) f'(\xi_x)\, dx \right| .
</math>

We can further approximate the integral on the right-hand side by bringing the absolute value into the integrand, and replacing the term in <math> f' </math> by an upper bound
{{NumBlk|:|
<math>
  \left| \int_a^b f(x)\, dx - (b - a) f(a) \right| 
  \leq {(b - a)^2 \over 2} \sup_{a \leq x \leq b} \left| f'(x) \right| ,
</math>
|{{EquationRef|1}}
}}
where the [[supremum]] was used to approximate.

Hence, if we approximate the integral <math display="inline"> \int_a^b f(x) \, dx </math> by the [[#Methods for one-dimensional integrals|quadrature rule]] <math> (b - a) f(a) </math> our error is no greater than the right hand side of {{EquationNote|1}}. We can convert this into an error analysis for the [[Riemann sum#Definition|Riemann sum]], giving an upper bound of
<math display="block">\frac{n^{-1}}{2} \sup_{0 \leq x \leq 1} \left| f'(x) \right|</math>
for the error term of that particular approximation. (Note that this is precisely the error we calculated for the example <math>f(x) = x</math>.) 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 proof]]s and ''verified'' calculations.