Approximate Calculation of Integrals

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 1.1 $$\int_{a}^{b}{p\left( x \right)f\left( x \right)dx\,\cong \,\sum\limits_{k\,=\,1}^{n}{A_{k}^{\left( n \right)}f\left( x_{k}^{\left( n \right)} \right)}}$$ 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)}\,