Editing Numerical integration (section)

== Connection with differential equations ==
The problem of evaluating the definite integral
:<math>F(x) = \int_a^x f(u)\, du</math>
can be reduced to an [[initial value problem]] for an [[ordinary differential equation]] by applying the first part of the [[fundamental theorem of calculus]]. By differentiating both sides of the above with respect to the argument ''x'', it is seen that the function ''F'' satisfies
:<math> \frac{d F(x)}{d x} = f(x), \quad F(a) = 0. </math>

[[Numerical methods for ordinary differential equations]], such as [[Runge–Kutta methods]], can be applied to the restated problem and thus be used to evaluate the integral. For instance, the standard fourth-order Runge–Kutta method applied to the differential equation yields Simpson's rule from above.

The differential equation <math>F'(x) = f(x)</math> has a special form: the right-hand side contains only the independent variable (here <math>x</math>) and not the dependent variable (here <math>F</math>). This simplifies the theory and algorithms considerably. The problem of evaluating integrals is thus best studied in its own right.

Conversely, the term "quadrature" may also be used for the solution of differential equations: "[[Linear_differential_equation#Types_of_solution|solving by quadrature]]" or "[[Ordinary_differential_equation#Reduction_to_quadratures|reduction to quadrature]]" means expressing its solution in terms of [[antiderivative|integrals]].