Welcome to Week 5! Having successfully concluded the section on Approximation Theory, this week we embark on a new topic: Numerical Integration, also known as Quadrature. Most integrals that arise in nature cannot be computed in closed form, or even evaluated exactly, and need to be approximated numerically. As integrals are limits of sums, a typical numerical integration procedure will consist in adding up weighted function values at carefully chosen points. In the first two lectures on this topic, we will motivate the general problem with an example, introduce the necessary terminology and concepts, recall some special cases from Numerical Analysis 1 (Trapezium rule), and study the midpoint rule.

Intended Learning Outcomes

At the end of the week you should be able to:

  • explain, using an example, the importance of numerical integration
  • apply the Trapezium and subdivided trapezium rule
  • derive the error bound for the midpoint rule

Tasks and Materials

  • Read through the preliminaries document, in particular the various forms of the Mean Value Theorem! These will be used repeatedly when analysing the error of quadrature rules.
  • Lecture 9 will be available as video recorcing on the Lectures page.
  • The fourth problem sheet is available and will be discussed this week, please have a look at it before class! Solutions to the previous problem sheet have been provided.