Calculation of definite integrals by the fundamental formula of integral calculus, 

$$\int_{a}^{b}{f\left( x \right)dx\,=\,F\left( b \right)\,-\,F\left( a \right)},$$

where f(x), let us say, is a continuous function on [a, b] and F(x) is its primitive function, is made difficult by the fact that the actual determination of F(x) is possible only in rare cases. For this reason, formulas for the approximate calculation of integrals are of great significance. In this chapter, we shall become acquainted with the most important of them. Many formulas for the approximate calculation of definite integrals have the form 

$$\int_{a}^{b}{p\left( x \right)f\left( x \right)dx\,\cong \,\sum\limits_{k\,=\,1}^{n}{A_{k}^{\left( n \right)}f\left( x_{k}^{\left( n \right)} \right)}}$$

(1.1)

and are called mechanical quadrature formulas. The sum on the right hand side of (1.1) is called the quadrature sum. The numbers x k (n) belong to the interval [a, b], and are called the knots of the quadrature formula, and the numbers A k (n) are the coefficients of the quadrature formula. We shall always consider the knots of the quadrature formula to be numbered in increasing order: 

$$x_{1}^{\left( n \right)}\,<\,x_{2}^{\left( n \right)}\,<\,\ldots \,<\,x_{n}^{\left( n \right)}.$$

The interval of integration [a, b] can also be infinite. The integrand is written in the form of the product of the two functions p(x) and f(x). The first of these, p(x), is assumed to be fixed for the given formula (1.1) and is called the weight function. The function f(x) belongs to some sufficiently wide class of functions, for example, continuous and such that the integral on the left hand side of (1.1) exists.

A. W. P. Canton

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