Week 5: Improper Integrals, and a first look at the Real Numbers
Welcome to Week 5. We have two themes this week. On Wednesday we will conclude our work on integral calculus by dicussing the topic of improper integrals. These resemble definite integrals, except that the regions whose areas are involved are not necessarily bounded – because they extend over an infinite region of the horizontal axis, or because some vertical asymptotes of the relevant function occur between the limits of integration. We may be looking at a region that has infinite width, but height that approaches zero as we travel along, and asking whether it is possible that the area enclosed could be finite. Our study of integral calculus gives us a context for making sense of this question and answering it in many cases. We will talk a little bit about what to expect in the exam relating to Chapter 1.
The topic of improper integrals is a good prelude to our work in Chapter 2, which will be about special properties of the real numbers (that distniguish them for example from the rational numbers) and about how we think about infinity and infinite sets. We will make a start on this topic in Thursday’s lecture.
Relevant sections of the lecture notes this week are Section 1.5 (from page 29) in Chapter 1 and Section 2.1 in Chapter 2. There are more examples in the lecture notes than we will discuss in our lectures, and more detailed explanations in some places.
Here is a pre-recorded version of Lecture 9, on improper integrals.
An interesting related topic is Gabriel’s Horn. As we saw in the lecture, the graph y=1/x encloses an infinite area over the region of the positive X-axis to the right of 1. Gabriel’s Horn is the solid object obtained by rotating this region around the X-axis. Remarkably, it has finite volume but infinite surface area (in our lecture today we showed that its vertical cross-section through the X-axis has infinite area, yet the entire object has finite volume which is curious and strikes many people as counter-intuitive). The wiki page linked above has nice explanations and is worth a look if you are interested in this.
Here is a pre-recorded version of Lecture 10, on the real numbers.
Supplementary video tutorial with more examples
Some more “typical” examples, on improper integrals. Notes from this tutorial.
Weekly Problem 5
The weekly problems are just for fun. They have nothing much to do with our curriculum. Please send me an email if you have a solution that you would like to share with the class!