Editing Integral (section)

===Numerical===
{{Main|Numerical integration}}
[[File:Numerical_quadrature_4up.png|right|thumb|Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature]]
Definite integrals may be approximated using several methods of [[numerical integration]]. The [[rectangle method]] relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the [[trapezoidal rule]], replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=519–520}}.</ref> The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further: [[Simpson's rule]] approximates the integrand by a piecewise quadratic function.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=522–524}}.</ref>

Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the [[Newton–Cotes formulas]]. The degree {{mvar|n}} Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree ''{{mvar|n}}'' polynomial. This polynomial is chosen to interpolate the values of the function on the interval.<ref>{{Harvnb|Kahaner|Moler|Nash|1989|p=144}}.</ref> Higher degree Newton–Cotes approximations can be more accurate, but they require more function evaluations, and they can suffer from numerical inaccuracy due to [[Runge's phenomenon]]. One solution to this problem is [[Clenshaw–Curtis quadrature]], in which the integrand is approximated by expanding it in terms of [[Chebyshev polynomials]].

[[Romberg's method]] halves the step widths incrementally, giving trapezoid approximations denoted by {{math|''T''(''h''<sub>0</sub>)}}, {{Math|''T''(''h''<sub>1</sub>)}}, and so on, where {{math|''h''<sub>''k''+1</sub>}} is half of {{math|''h''<sub>''k''</sub>}}. For each new step size, only half the new function values need to be computed; the others carry over from the previous size. It then [[Interpolation|interpolate]] a polynomial through the approximations, and extrapolate to {{math|''T''(0)}}. [[Gaussian quadrature]] evaluates the function at the roots of a set of [[orthogonal polynomials]].<ref>{{Harvnb|Kahaner|Moler|Nash|1989|p=147}}.</ref> An {{mvar|n}}-point Gaussian method is exact for polynomials of degree up to {{math|2''n'' − 1}}.

The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as [[Monte Carlo integration]].<ref>{{Harvnb|Kahaner|Moler|Nash|1989|pp=139–140}}.</ref>