Numerical Integration And Integration Rules Based On Uniform Mesh Spacing
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(xk)’s are called the ordinates and λk’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 Rn = 0, for all polynomials of degree less than or equal to p. That is, it produces exact results for f(x) = 1, x, x2, ...., x p. This implies that
The error term is obtained for f(x) = xp 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 Rn(xp 1) also becomes zero, then the error term is obtained for f(x) = xp 2.
Integration Rules Based on Uniform Mesh Spacing :
When w(x) = 1 and the nodes xk’s are prescribed and are equispaced with x0 = a, xn = b, where h = (b – a)/n, the methods (1.2) are called Newton-Cotes integration rules. The weights λk’s are called Cotes numbers.
where 
defines the area under the curve y = f(x),