Month: November 2023 (Page 1 of 3)

Class Info

• Date: Mon Nov 20 & Mon Nov 27
• Meeting Info: 10a-11:40a, N719

Announcements

WebWork:

• “Series – Infinite Series” – due Fri Dec 1
• additional WebWorks on infinite series will be assigned as we cover the topics in class, so please keep up with the WebWork!

Topics

On Monday Nov 20, we went over Exam #2, and then introduced the last topic we will cover this semester–infinite series, i.e., infinite sums. We introduced “Sigma” notation, partial sums, and the idea of convergence/divergence (covered in Sec 5.2 of the textbook):

We made a connection to improper integrals–this is the “integral test” for convergence/divergence:

We went through some of the exercises from the WebWork, introducing partial sums:

On Mon Nov 27, we continued with these topics:

We talked about how convergence of an infinite series is defined in terms of the partial sums:

We then discussed geometric series:

Here is the passage from the textbook that defines geometric series, with an example:

Class Info

• Date: Mon Nov 13
• Meeting Info: 10a-11:40a, N719

Announcements

Exam #2:

• Our 2nd midterm exam will be tomorrow, Wednesday Nov 15
• See this post for a list of topics and review exercises for the exam
• Please work through on those review exercises, which I will collect later in the semester as a HW assignment

Topics

We reviewed the topics that will be on the exam, in particular partial fractions and volumes of revolution. We reviewed Quiz #3, and set up a couple of the suggested exercises from the Final Exam Review sheet:

We also revisited the topic of improper integrals, although this will not be on tomorrow’s exam:

We closed with an application of improper integrals to “an infinite” volume of revolution–“Gabriel’s horn“:

Class Info

• Date: Wed Nov 8
• Meeting Info: 10a-11:40a, N719

Announcements

WebWork:

• “Applications – Volumes of Revolution” (due Friday Nov 9)

Exam #2:

• Our 2nd midterm exam will be Wednesday (Nov 14)
• See this post for a list of topics and review exercises for the exam
• Please start working on those review exercises, which I will collect later in the semester as a HW assignment

Topics

We reviewed the last exercise on the “Volumes of Revolution” WebWork, which asked for the volume of a torus:

We found that setting up an integral for this volume is rather complicated, but in fact it’s quite straightforward to find the volume if we “cut” the torus and “unwrap” it–so that we see that it has the same volume as a cylinder. Here is an illustration (from this website) of what we discussed, and what I tried to draw on the board:

We then did some additional examples of improper integrals. We revisited the improper integral we calculated on Monday, which “converges” to a finite number, even though we are integrating over an infinite interval:

You can also see that example in this PatrickJMT video:

We contrasted this with the improper integral of 1/x over the same interval, which we saw diverges to infinity. That example is worked out in Sec 3.7 of the textbook:

You should also read Example 3.48, which is an improper integral for a volume of revolution via the disk method!