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!

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