## Numerical Integration

**Numerical integration **is the approximate computation of an integral using numerical techniques. In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.The problem of numerical integration is to find an approximate value of the integral

.....................1.1

where w(x) > 0 in (a, b) is called the weight function. The function f(x) may be given explicitly or as a tabulated data. We assume that w(x) and w(x) f(x) are integrable on [a, b]. The limits of integration may be finite, semi-infinite or infinite. The integral is approximated by a linear combination of the values of f(x) at the tabular points as,

.....................1.2

The tabulated points xk’s are called abscissas, f(x_{k})’s are called the ordinates and λ_{k}’s are called the weights of the integration rule or quadrature formula (1.2). We define the error of approximation for a given method as

............................1.3

Order of a method An integration method of the form (1.2) is said to be of order p, if it produces exact results, that is R_{n} = 0, for all polynomials of degree less than or equal to p. That is, it produces exact results for f(x) = 1, x, x^{2}, ...., x ^{p}. This implies that

The error term is obtained for f(x) = x^{p 1}. We define

..............................1.4

where c is called the error constant. Then, the error term is given by

.......................1.5

The bound for the error term is given by

............................1.6

If R_{n}(x^{p 1}) also becomes zero, then the error term is obtained for f(x) = x^{p 2}.

**Integration Rules Based on Uniform Mesh Spacing :
**

When w(x) = 1 and the nodes x_{k}’s are prescribed and are equispaced with x_{0} = a, x_{n} = b, where h = (b – a)/n, the methods (1.2) are called Newton-Cotes integration rules. The weights λ_{k}’s are called Cotes numbers.

where defines the area under the curve y = f(x),