Category: Class Agendas (Page 4 of 9)

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):

Here is a better sketch for that exercise:

And here is an illustration of the general case, and the resulting integral:

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:

We set up an example from the WebWork:

Here is a video with more examples using the cylindrical shells method:

Class Info

• Date: Mon Oct 30
• Meeting Info: 10a-11:40a, N719

Announcements

WebWork:

• finish “Integration – Partial Fractions” (due tonight – Mon Oct 30)
• “Applications – Areas Between Curves” (due Fri Nov 3)
• “Applications – Volumes of Revolution” (due Nov 8)

Topics

We went through some examples of computing “volumes by slicing” (covered in Sec 2.2 of the textbook) which also called “the disk method”–since we form the definite integral for the volume by picturing the object “sliced” into thin disks.

We first calculated the volume of a cylinder and the volume of a cone using this method. In the first case, the “cross-sectional” area is a constant; for a cone, the cross-sectional area is a circle whose radius is a linear function of x:

This leads to the general form of the definite integral for calculating a volume by slicing, in terms of the cross-sectional area; and the particular form of that for a “solid of revolution”:

We set up Example 2.7 from Sec 2.2 of the textbook:

We then went through the first few exercises from the “Applications – Volumes of Revolution” WebWork set. Here is #3:

Here is a video that illustrates the disk method, and a variation called “the washer method”:

Some of the subsequent exercises in that WebWork set use a different technique for computing volumes, called “cylindrical shells.” We will discuss that topic on Wednesday.

Class Info

• Date: Wed Oct 25
• Meeting Info: 10a-11:40a, N719

Announcements

WebWork:

• finish “Integration – Partial Fractions” (extended to Mon Oct 30)
• work on “Applications – Areas Between Curves” (due Wed Nov 1)

Topics

We went through some examples of computing “areas between curves” from the WebWork. This the first of some applications of integration we will study:

Here is the example we set up in class Monday:

As illustrated above, the general strategy involves identifying which curve (i.e., function) is “the top boundary” of the area and which is the “bottom boundary.” And in many examples (such as the one above) we have to solve for where the curves intersect, in order to figure out the endpoints of the interval of integration:

Here is a more complicated example we set up in class (Problem 3 on the WebWork). The two curves intersect at 3 points, with which curve is on top switching at the middle point. So we need to set up two separate integrals for the total area enclosed:

We also discussed an example (Problem 5) where it is more convenient to integrate with respect to y:

Here is the video I posted in the previous Class Recap:

And here is a followup video showing an example of integrating with respect to y: