Category: Pre-class prep lessons (Page 1 of 2)

2.3 Volumes of Revolution: Cylindrical Shells 

P. 166: 120 – 131 all, 140-143 all, 145, 148, 158, 159
P. 271: 61

Webwork: Shells and Washers due 5/17

Notes from lecture and office hours:

https://www.dropbox.com/s/5s7xtq2y6aknag9/Note%20May%2013%2C%202020.pdf?dl=0

Method of cylindrical shells

It will be helpful to remember the pictures from Monday’s lecture.

The method of cylindrical shells is another method for calculating volumes of solids of revolution. It differs from the methods discussed previously in that we’re no longer slicing to get areas of cross sections. (In some sense, though, we still are slicing; but the slices are no longer flat, they’re cylindrical). But the general principal is the same: we’ll integrate 2-dimensional areas to get the 3-dimensional volume.

Remember that vertical segment connecting the $x$-axis with the graph $y=f(x)$ at position $x$? When we rotated it around the $x$-axis, we got a disk. But what shape would we get if we rotate it around the $y$-axis (a line that’s parallel to the vertical segment instead of perpendicular to it)?

The visualization might take a bit of time if you haven’t tried to see it before. Hold your pencil vertically and then move your hand in a circle that’s parallel to the floor. Imagine you took a stop-motion video of this; what shape would your pencil sweep out?

Well, this method is called “cylindrical shells” because the answer to both of these questions is “a cylindrical shell.” Scroll to Figure 2.26 here. Your cylinder is hollow and has no top or bottom. The area we’re interested in is its surface area. Try cutting this cylinder vertically; what shape do you get?

How can you understand the formula:

$V = \int_a^b 2 \pi f(x) dx$?

• This video (8 minutes) takes you through the formula and shows an example where the axis of rotation is the $y$-axis.
• This video (9 minutes) does the same thing, but now the axis of rotation is the $x$-axis.  This approach tends to be helpful if your graph gives $x$ in terms of $y$: $x = f(y)$.

Which volume method when?

This is the big question. Part of the challenge in calculating volumes is knowing when to use which method. Following these segments through a rotation will help you determine the shape you’re dealing with; it will be a disk, a washer, or a cylindrical shell.

2.2 Determining Volumes by Slicing (part 2)

P. 150: 58, 59, 74 – 80 all, 98 – 102 all
Find the volume of the solid obtained by rotating the region bounded by the curves y = x^2, y = 12-x, x = 0 and x ≥ 0 about (a) the x–axis; (b) the line y = -2; (c) the line y = 15; (d) the y-axis; (e) the line x = -5; (f) the line x = 7.

Webwork: Shells and Washers due 5/17

Lecture notes: https://www.dropbox.com/s/lugsw916xjk5j3l/Note%20May%2011%2C%202020.pdf?dl=0

Back to volumes by slicing

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2.2 Determining Volumes by Slicing (p. 141 – 149)

• P. 150: 58, 59, 74 – 80 all, 98 – 102 all
• Find the volume of the solid obtained by rotating the region bounded by the curves y = x^2, y = 12-x, x = 0 and x ≥ 0 about
  • (a) the x–axis;
  • (b) the line y = -2;
  • (c) the line y = 15;
  • (d) the y-axis;
  • (e) the line x = -5;
  • (f) the line x = 7.

Webwork:

• Volumes (optional) due 5/12
• Shells and washers (for now, look for volumes that can be calculated by the disk method) due 5/17

Motivation

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1.1 Approximating Areas (p. 5 – 20)
2.1 Areas Between Two Curves (p. 122 – 128)

P. 21: 1 – 7 odd, 12, 15, 16, 17 (optional)
P. 131: 1 – 7 all, 11, 15 – 21 all, 23
P. 271: 63

Webwork: Riemann Sums (optional), Area Between Curves due 5/10

Notes from lecture and office hours:

https://www.dropbox.com/s/z6bo23k3staxnpm/Note%20May%204%2C%202020.pdf?dl=0

1.1. Riemann Sums

The Webwork set for Riemann Sums is officially optional, but the pictures in this section will help set us up to understand how integrals can be used to compute volumes, which is the last big topic in the course. Continue reading

6.3 Taylor and Maclaurin Series (p.561–562, 573–576)
6.4 Working with Taylor Series (p.584–587, 590–592)

P. 578: 118-123 all, 140—147 all, 151—155 all
P. 596: 203, 206, 207, 209, 219–223 odd

Webwork Taylor series due Tuesday, May 5

Notes from lecture and office hours: https://www.dropbox.com/s/f8gxpteui7easip/Note%20Apr%2029%2C%202020.pdf?dl=0

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6.1 Power Series and Functions (p.531–537)
6.2 Properties of Power Series (p.544–548, 552–557)

P. 541: 13-21 odd, 24, 28
P. 558: 87—90 all, 96, 97

Webwork: Power Series due Sunday, May 3

Lecture notes:

https://www.dropbox.com/s/i86orzrwnoc15s8/Note%20Apr%2027%2C%202020.pdf?dl=0

Sorry, no video again today! I’ll see if I can figure Webwex out…

These two sections are scheduled for one day, because they’re really about different aspects of one topic: power series. Continue reading

5.5 Alternating Series (p.496–502) HW: P. 505: 250–257 all, 261—264 all, 266, 267

5.6 Ratio and Root Tests (p.509–519) HW: P. 522: 317–320 all, 323, 325, 328, 329–335 odd, 349, 351

Webwork: Alternating Series & Ratio and Root Tests due Tuesday

Notes from lecture and office hours: https://www.dropbox.com/s/i58xxrwn4ioapmb/Note%20Apr%2022%2C%202020.pdf?dl=0

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5.4 Comparison Tests (p.485–492)

P. 493: 194—197all, 199, 200, 202, 204—206 all, 211 (optional: 222-223)

Webwork : Comparison Tests due Sunday 4/26

Notes: https://www.dropbox.com/s/58bi682wpfinnjl/Note%20Apr%2020%2C%202020.pdf?dl=0

Video: https://www.dropbox.com/s/2kqcysxe0ok5g2p/Lecture%20%26%20Office%20Hours-20200420%201228-1.mp4?dl=0

5.3 The Divergence and Integral Tests

Textbook PDF pp.471-478

Textbook HW: p. 482: 138–145 odd, 152—155, 158, 159, 161, 163

Webwork: Integral Test and Divergence Test both sets due 4/21 (these two Webwork sets are relatively short, so don’t forget to practice the textbook homework as well).

Notes from lecture and office hours

Video:

https://www.dropbox.com/s/rlz1wsvktafqjzm/Lecture%20%26%20Office%20Hours-20200415%201224-1.mp4?dl=0

Divergence Test

You saw the divergence test briefly in Monday’s lesson.

The divergence test is convenient when it applies. It’s always a good idea to check whether it applies to a particular series before you start trying to use another test for convergence.

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5.2 Infinite Series

Textbook PDF pp. 450–459

Textbook HW: p. 466: 67–74, 76, 77, 79, 80, 83–85 odd, 89—95 odd

Webwork: Intro to Series due 4/19

Notes from lecture and office hours: Note Apr 13, 2020

Video from lecture and office hours https://www.dropbox.com/s/dimiyy58j0p36tn/Lecture%20%26%20Office%20Hours-20200413%201207-1.mp4?dl=0

Motivation

Remember that our ultimate goal this chapter is to understand what it means for an “infinite degree Taylor polynomial” to “represent” a function near a point. We said that our first step toward this is to understand what it means for a sequence to converge, how to determine if a sequence converges, and (if a sequence does converge) determine what it converges to. Continue reading