4.3 Elements of Numerical Integration The need often arises for evaluating the definite integral of a function that has no explicit antiderivative or whose antiderivative is not easy to obtain. The basic method involved in approximating is called numerical quadrature. It uses a sum to approximate .

Basic Idea We first select a set of distinct nodes from the interval . Then we integrate the Lagrange interpolating polynomial and its truncation error term over to obtain The quadrature formula is, therefore, with error given by

Before discussing the general situation of quadrature formulas, let us consider formulas produced by using first and second Lagrange polynomials with equally spaced nodes. This gives the Trapezoidal rule and Simpson’s rule, which are commonly introduced in calculus courses.

the Trapezoidal rule To derive the Trapezoidal rule for approximating let and use the linear Lagrange polynomial then, Apply theWeighted Mean Value Theorem for Integrals to the error term

the Trapezoidal rule Thus, we have Since the error term for the Trapezoidral rule involves , the rule gives the exact result when applied to any function whose second derivative is identically zero, that is, any polynomial of degree one or less.

Simpson’s rule Simpson’s rule results from integrating over the second Lagrange polynomial with nodes and ,where . Therefore, .

Simpson’s rule Simpson’s Rule: Since the error term involves the fourth derivative of , Simpson’s rule gives exact results when applied to any polynomial of degree three or less.

Degree of accuracy Definition 4.1 The degree of accuracy, or precision, of a quadrature formula is the largest positive integer n such that the formula is exact for Definition 4.1 implies that Trapezoidal and Simpson’s rules have degree of precision one and three, respectively.

Example 1 The quadrature formula is exact for all polynomials of degree less than or equal to 2. Determine the constants A,B,C. Solution. Let f(x)=1,x,x2 ,we obtain a system of linear equations Thus,

Example 2 Find the constants A-1, A0 ,A1 so that the quadrature formula has the highest possible degree of precision and find its degree of precision 。 解：令求积公式对f(x)=1, x, x2准确成立，则有

Example 2 Find the constants A-1, A0 ,A1 so that the quadrature formula has the highest possible degree of precision and find its degree of precision 。 解之得 其代数精度至少为2,将f(x)=x3代入求积公式两端相等,而将将f(x)=x4代入求积公式两端不相等,所以其代数精度为3次 .

Closed Newton-Cotes formula The Trapezoidal and Simpson’s rules are examples of a class of methods known as Newton-Cotes formulas. There are tow types of Newton-Cotes formulas, open and closed. The (n+1)-point closed Newton-Cotes formula uses nodes where and It is called closed because the endpoints of the closed interval [a,b] are included as nodes.

Theorem 4.2 For the (n+1)-point closed Newton-Cotes formula

Open Newton-Cotes formula The open Newton-Cotes formulas used nodes where This implies that so we label the endpoints by setting Open formulas contain all the nodes used for the approximation within the open interval (a,b). The formulas become n=0: Midpoint rule