1. 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

2. 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

3. 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.

4. 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

5. 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.

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

7. 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.

8. 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.

9. 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,

10. 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

11. 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 .

12. 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.

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

15. 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