Connection With Differential Equations
The problem of evaluating the integral
can be reduced to an initial value problem for an ordinary differential equation. If the above integral is denoted by I(b), then the function I satisfies
Methods developed 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 I ' (x) = ƒ(x) has a special form: the right-hand side contains only the dependent variable (here x) and not the independent variable (here I). This simplifies the theory and algorithms considerably. The problem of evaluating integrals is thus best studied in its own right.
Read more about this topic: Numerical Integration (quadrature)
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