Class Info
- Date: Wed Nov 1
- Meeting Info: 10a-11:40a, N719
Announcements
WebWork:
- “Applications – Areas Between Curves” (due Sun Nov 5)
- “Applications – Volumes of Revolution” (due Friday Nov 9)
We will have a quiz on Monday, with one exercise on integrating using the method of partial fractions and one exercise on computing the area between two curves.
Topics
We went through a few additional examples from the “Volumes of Revolution” WebWork set.
We introduced “the washer method,” which variation on the disk method, also used for calculating the volume of a solid of revolution “by slicing”–but for solids which have a hollow “cavity”, and so the object is sliced into “washers” (disks with an empty inner circle):
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/20231101_112044-1024x768.jpg)
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/20231101_113654-1024x768.jpg)
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/20231101_114809-1024x768.jpg)
Here is a better sketch for that exercise:
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/180004.image1_.jpg)
And here is an illustration of the general case, and the resulting integral:
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/mahima-06-april-disk-method-formula-05-1618413195-1024x764.png)
See also Example 2.10 from Sec 2.2 of the textbook. We watched parts of this video as well:
We also introduced a different method for calculating the volume of a solid of revolution: the method of “cylindrical shells.”
This is covered in Sec 2.3 of the textbook. Here are the images we looked at:
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/Screen-Shot-2023-11-02-at-2.49.56-PM.png)
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/Screen-Shot-2023-11-02-at-2.50.06-PM.png)
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/Screen-Shot-2023-11-02-at-2.50.23-PM.png)
We set up an example from the WebWork:
![](https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/files/2023/11/20231101_114831-1024x768.jpg)
Here is a video with more examples using the cylindrical shells method:
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