Chapter 8: Numerical Integration Topics

• Know what integration is (Section 8.1)
• Know what an n-point quadrature rule is (Section 8.3)
• Know how Lagrange basis functions are used to compute the weights for Newton-Cotes quadrature. (Description in 8.3, examples in class notes)
• Understand the method of undetermined coefficients for computing weights (mentioned in class, example in 8.3)
• Know the difference between open and closed Newton-Cotes rules
• Be able to use Taylor series to derive error estimates for various rules (note plus pages 347 - 348). Understand the idea of using two different approximations of an integral to approximate error. For example, using Taylor series, we derived the following:
  • the error in the midpoint rule can be approximated using the midpoint and the trapeziod approximations of the integral (know the approximation)
  • a relationship between the error in the midpoint rule and the trapezoid rule (so together with the above you could use the midpoint rule and the trapezoid rule to approximate the error in the trapeziod rule) -- know the relationship!
  • the error in approximating an integral using Simpson's rule with 2 subdivisions of an interval can be approximated using the 2 subdivision approximation and the 1 subdivision approximation of the integral (know the error approximation)
• Know what is meant by polynomial degree of a quadrature rule (page 344)
• Know what a composite method is. Know the order of the error for Midpoint, Trapezoid, and Simpson composite rules. (pp. 355 - 356)
• Understand the idea of adaptive rules. Understand why such rules are used (what they accomplish). Be able to illustrate how such rules would subdivide the interval for a given graph. In particular understand Adaptive Simpson. (pp. 356 - 359)
• Understand the Monte Carlo method for approximating an integral. Why is this method not recommended for one-dimensional integrals but is good for higher dimensions? (Class plus brief discussion in Section 8.4.4 page 361 - 362)

Review Questions: (pages 373-375) #8.2, 8.3, 8.8, 8.14, 8.18, 8.27, 8.28, 8.33
Exercises: (pages 376 - 377) #8.1, 8.2(a), 8.3, 8.4, 8.6, 8.11(a)