175- Disk- vs. Shell method

August 28, 2008

Here is the computation I wanted to show at the end of today’s lecture: Consider a function f(x) that is continuous and invertible on an interval \null[a,b]. Rotate about the y-axis the region bounded by y=f(a), x=b, and y=f(x). Then the values assigned to the volume of the resulting solid by applying either the disk method or the shell method coincide.

To see this, let’s call D the expression obtained using the disk (or washer) method and S the expression obtained using the shell method.

Then D=\int_{f(a)}^{f(b)} \pi[b^2-(f^{-1}(y))^2]\,dy and S=\int_a^b 2\pi x(f(x)-f(a))\,dx.

As I mentioned in lecture, the idea to show the equality of both expressions is to, more generally, consider functions D(t) and S(t) that compute the values given by the disk and shell methods, respectively, for the volume of the solid obtained by rotating about the y-axis the region bounded by y=f(a), y=f(x), and x=t. Here, t is a new variable that varies in \null[a,b]. Hence, D(t)=\int_{f(a)}^{f(t)} \pi[t^2-(f^{-1}(y))^2]\,dy and S(t)=\int_a^t 2\pi x(f(x)-f(a))\,dx. We will show that D(t)=S(t) for all t\in[a,b].

The trick is to proceed indirectly and, rather than looking at D(t) and S(t), we examine their derivatives. Of course, if D(t)=S(t) for all t\in[a,b], then we must also have D'(t)=S'(t). Let’s check that this is indeed the case:

We have S'(t)=2\pi t (f(t)-f(a)). To compute D'(t), let’s simplify D(t) a bit before differentiating: D(t)=\int_{f(a)}^{f(t)} \pi t^2\,dy-\int_{f(a)}^{f(t)}\pi f^{-1}(y)^2\,dy, so D(t)=\pi t^2(f(t)-f(a))-\pi\int_{f(a)}^{f(t)} f^{-1}(y)^2\,dy (Notice that t does not depend on y, so we can take t^2 out of the integral sign.)

Then D'(t)=2\pi t(f(t)-f(a))+\pi t^2 f'(t)-(\pi\int_{f(a)}^{f(t)} f^{-1}(y)^2\,dy)' so D'(t)=2\pi t(f(t)-f(a))+\pi t^2 f'(t)- \pi f^{-1}(f(t))^2f'(t), or D'(t)=2\pi t(f(t)-f(a)), since f^{-1}(f(t))=t.

We have found that S'(t)=2\pi t (f(t)-f(a))=D'(t).

This does not automatically imply that S(t)=D(t), but almost: Let h(t)=S(t)-D(t). Then h'(t)=0 (for all t\in(a,b)), so h(t) is constant.

Finally, notice that h(a)=S(a)-D(a)=0-0=0. Since h is constant, it follows that S(t)=D(t) for all t\in[a,b]. And we are done.

Remark 1. We will see a different argument later in the course, once we study integration by parts.

Remark 2. One can also check that the disk and shell methods provide the same result when we rotate about the y-axis the region bounded by y=f(x), y=f(b) and x=a. To check your understanding of the argument above, it may be useful to try to work this case out on your own; the algebra is somewhat simpler than the computations I just detailed.

Leave a Comment » | 175: Calculus II | Permalink
Posted by andrescaicedo

275 -Syllabus

August 24, 2008

Math 275: Calculus III.


Instructor: Andres Caicedo.
Time: MTWF 9:40-10:30am.
Place: Mathematics/Geosciences building, Room 139.

Text: Hass, Weir, Thomas, University Calculus.

Contents: Chapters 10-14. I will frequently update this entry with more detailed week to week descriptions; if time allows it, additional topics than indicated may be covered.

  • August 25-29: Review of Calculus I and II, Chapter 10 up to section 10.5.
  • September 2-5: Remainder of section 10.5, 10.6, and beginning of 11.1.
  • September 8-12: Sections 11.1-4.
  • September 15-19: Remainder of Chapter 11.
  • September 29-October 3:  Chapter 12 up to section 12.3.
  • October 6-10: Remainder of 12.3, 12.4, 12.5,  and beginning of 12.6. 
  • October 13-17: Sections 12.5-12.7 and beginning of 12.8.
  • October 20-24: We will skip section 12.8. Sections 12.7, 12.9, 13.1.
  • November 3-7: Chapter 13 until 13.7 if possible. `Mean value property’ of harmonic functions.
  • November 10-14: Sections 13.6-13.8. General version of the Chain rule.
  • November 17-21: Sections 14.1-14.4.
  • December 1-5: Sections 14.3-14.6.

Prerequisites: 175 (Calculus II) or equivalent. Linear algebra would be desired but it is not required.

Exams: There will be 2 in-class exams and a comprehensive final exam.

  • Exam 1: Friday September 26.
  • Exam 2: Friday October 31.
  • Final exam: Monday December 15, 10:30am-12:30pm.

Dates and times are non-negotiable. Failure to take a exam will be graded as a score of 0. There will be no make up for the final exam. For the in-class exams, a make up can be arranged if I am notified prior to the exam date and a valid reason is presented; keep in mind that make up exams will be more difficult than regular in-class exams.

Homework: There is weekly homework, due Tuesdays at the beginning of the class. I will frequently update this entry with each week’s homework assignment. No late homework will be graded. Failure to turn in a homework set corresponds to a score of 0. The lowest homework score will be dropped at the end of the term. Some homework sets will have a small amount of extra credit points.

  • Homework 1, due Wednesday, September 3. Solve the following exercises: From section 10.2, #49, 53. From 10.3, #17, 18. From 10.4, #28. From 10.5, #22. Please explain (and show) all your work. Each exercise is worth 2 points. The set is worth 10 points, and you can get 2 extra credit points.
  • Homework 2, due Tuesday, September 9. From section 10.5, solve exercises #45, 46, 64, 66. From section 10.6, exercises #1-12. From the practice exercises for chapter 10, #62. From the additional and advanced exercises for chapter 10, #18. Each exercise from section 10.5 is worth 1 point. The combined question from section 10.6 is worth 2 points. The other two exercises are worth 2 points each, for a total of 10 points. 
  • Homework 3, due Tuesday, September 16. From section 11.1, exercises #20, 24. From 11.2, #14, 26. From 11.3, #14. From 11.4, #22. The set is worth 10 points, each exercise is worth 2 points, and you can get 2 extra credit points. Please turn in your homework at the beginning of lecture, not in the middle of lecture or at the end, not when you arrive to lecture if you arrive late. Starting with this homework set, I’ll be getting pickier on how you write your solutions. Please write them in a reasonable fashion so the person reading them can understand what you are doing without having to go check in the book what formula you are trying to use. Use words if necessary. A long string of equations and implications is not particularly readable, unless the reader already knows what you are trying to do, so try to make your solutions a bit more reader-friendly than they have been so far. This will also help you study later. You may lose points even if you have found the correct answer to an exercise but it is not written appropriately.
  • Homework 4, due Tuesday, September 23. Same remarks as above apply. From section 11.5, solve exercises #8, 18, 26. From 11.6, #4. From the practice exercises for Chapter 11, #10. Each exercise is worth 2 points.   
  • Homework 5, due Tuesday, October 7. Same remarks as before apply. Section 12.1, exercises 13-18, 44. Section 12.2, exercises 36, 54. Section 12.3, exercises 21, 22, 30. The exercises in section 12.3 are worth 1 point each, the combined exercise 12.1.13-18 is worth 3 points, the other exercises are worth 2 points each; there are two extra credit points.
  • Homework 6, due Tuesday, October 14. Same remarks as before apply. Section 12.3, exercises 63, 66, 68, 74-77. Section 12.4, exercises 4, 10, 32, 42-44, 48, 50. This homework will be graded out of 10 points. Each exercise is worth 1 point. You can turn in as many exercises as you want. Indicate the ones you want to be extra credit problems. Of those, 2 will be chosen randomly to be graded, so you can have up to 2 extra credit points.
  • Homework 7, due Tuesday, October 21. Same remarks as before apply. Section 12.5, exercises 6, 11, 16, 24, 28. Section 12.6, exercises 2, 6, 14, 18, 40, 56, 57. This homework will be graded out of 10 points. Each exercise is worth 1 point. You may obtain two extra credit points.
  • Homework 8, due Tuesday, October 28. Same remarks as before apply. Section 12.7, exercises 2, 20, 36, 40, 44, 46. Section 12.9, exercises 4, 10, 12. Section 13.1, exercises 6, 24. This homework will be graded out of 10 points. Each exercise is worth 1 point. You may obtain one extra credit point.
  • Homework 9, due Tuesday, November 11. Same remarks as before apply. Section 13.2, exercises 5, 17, 39, 47, 54. Section 13.3, exercises 13, 16, 21. Section 13.4, exercise 27, 36. Each exercise is worth 1 point.
  • Homework 10, due Tuesday, November 18. The usual considerations apply. Section 13.5, exercises 4, 8, 19, 22, 30, 45. Section 13.6, exercises 6, 11, 24, 30. Section 13.7, exercises 13, 21, 37, 60, 78, 79. Section 13.8, exercises 1, 6, 16, 21. There are 20 exercises this week. Turn in at least 10. The remaining problems (at most 10) will be due together with a few additional exercises on December 2. Each exercise is worth 1 point.
  • Homework 11, due Tuesday, December 2. The usual considerations apply. Turn in the problems listed for Homework 10  that you still have pending. Section 14.1, exercises 1-8, 12, 16, 29. Section 14.2, exercises 6, 8, 34, 41. Section 14.3, exercises 2, 6, 12, 17, 20, 34, 38. Section 14.4, exercises 2, 8, 18, 31-35. The combined exercises 14.1.1-8 count as a single problem. Besides the problems pending from last week, there are 23 exercises. Turn in at least 10 of these. The remaining problems (at most 13) will be due together with a few additional exercises on December 9. Each exercise is worth 1 point.
  • Homework 12, due Tuesday, December 9. The usual considerations apply. Turn in the problems you still have pending. Each exercise is worth 1 point.

Grading:

  • Homework: 60%.
  • Exam 1: 10%.
  • Exam 2: 10%.
  • Final exam: 20%.

I will then grade on a linear scale:

  • If your final score is 90% or higher, you receive an A.
  • If it is between 80 and 89%, you receive a B.
  • If it is between 70 and 79%, you receive a C.
  • If it is between 60 and 69% you receive a D.
  • If it is 59% or lower, you receive an F.
  • There may be a small curve up if the distribution of scores warrants this. Plus and minus grades might be used for grades near the top or bottom of a grade range.

Attendance: Not required, but encouraged. I will use this website to post any additional information, and encourage you to use the comments feature, but I will not post here content covered in class. If you leave a comment, please use your full name, which will simplify my life filtering spam out.

2 Comments | 275: Calculus III | Tagged: , | Permalink
Posted by andrescaicedo

175 -Syllabus

August 24, 2008

Math 175 Section 5: Calculus II.


Instructor: Andres Caicedo.
Time: MTWF 7:40-8:30am. (Ugh!)
Place: Business building, Room 217.

Text: Hass, Weir, Thomas, University Calculus.

Contents: Chapters 6-9. I will frequently update this entry with more detailed week to week descriptions. My general plan is a bit ambitious, and leaves a few additional hours free at the end of the semester, where additional topics could be covered. These hours will also act as a buffer in case some topics require more time than originally intended.

  • August 25-29: Review of Calculus I, Sections 6.1, 6.2.
  • September 2-5: Sections 6.3, 6.4, and half of 6.5.
  • September 8-12: Remainder of Chapter 6, Sections 9.1 and half of 9.2. 
  • September 15-19: Remainder of section 9.2, sections 9.3, 9.4, and half of 9.5.
  • September 29-October 3: Chapter 7 up to 7.2.
  • October 6-10: Sections 7.3, 7.4, 7.6.
  • October 13-17: Sections 7.6, 7.7, 8.9 (only as it concerns to Taylor polynomials and the error terms of these approximations). The discussion of the error term in Simpson’s rule is based on the article “Simpson’s rule is exact for quintics” by Louis A. Talman, American Mathematical Monthly, vol 113 February 2006, pp. 144-155.
  • October 20-24: Sections 7.7 and 8.1.
  • November 3-7: Sections 8.1-4.
  • November 10-14: Sections 8.3-5.  
  • November 17-21: Sections 8.6-9.
  • December 1-5: Section 8.7-8.10. 
  • December 8-12: Uniform and pointwise convergence of power series, Wierstrass test, infinite products, additional material on the p-series \displaystyle \sum_{n=1}^\infty\frac1{n^p}. 

Prerequisites: 170 (Calculus I) or equivalent.

Exams: There will be 2 in-class exams and a comprehensive final exam.

  • Exam 1: Friday September 26. Should cover Chapters 6 and 9.
  • Exam 2: Friday October 31. Should cover Chapters 7 and 8 (up to half of section 8.7).
  • Final exam: Monday December 15, 8-10am.

Dates and times are non-negotiable. Failure to take a exam will be graded as a score of 0. There will be no make up for the final exam. For the in-class exams, a make up can be arranged if I am notified prior to the exam date and a valid reason is presented; keep in mind that make up exams will be more difficult than regular in-class exams.

Homework: There is weekly homework, due Tuesdays at the beginning of the class. I will frequently update this entry with each week’s homework assignment. No late homework will be graded. Failure to turn in a homework set corresponds to a score of 0. The lowest homework score will be dropped at the end of the term. Some homework sets will have a small amount of extra credit points.

  • Homework 1, due Wednesday September 3: From section 6.1, solve exercises 8, 15, 35. From section 6.2, solve exercises 14, 24. Please display all your work, not just the final answer, and explain what you are doing rather than making me guess what you are trying to do. Each exercise is worth 2 points.
  • Homework 2, due Tuesday September 9: From section 6.3, solve exercises 14, 30. From section 6.4, exercise 28. From section 6.5, exercises 4, 8, 13, 18, 32, 34. From the practice exercises for chapter 6 (beginning in page 444), exercise 24. Each exercise is worth 1 point.  
  • Homework 3, due Tuesday, September 16: From section 6.5, exercises #25, 36. From 6.6, #2, 33. From 6.7, #8, 30. From 9.1, #2, 26, 30, 34. From 9.2, #2. The set is worth 10 points, each exercise is worth 1 point, and you can get 1 extra credit point. Please turn in your homework at the beginning of lecture, not in the middle of lecture or at the end, not when you arrive to lecture if you arrive late. Starting with this homework set, I’ll be getting pickier on how you write your solutions. Please write them in a reasonable fashion so the person reading them can understand what you are doing without having to go check in the book what formula you are trying to use. Use words if necessary. A long string of equations and implications is not particularly readable, unless the reader already knows what you are trying to do, so try to make your solutions a bit more reader-friendly than they have been so far. This will also help you study later. You may lose points even if you have found the correct answer to an exercise but it is not written appropriately.
  • Homework 4, due Tuesday, September 23. Same remarks as above apply. From section 9.2, solve exercsies #12, 24. From 9.3, #23, 30. From 9.4, #26, 44, 69. From 9.5, #38. From the practice section of Chapter 9, #17-24. Each exercise is worth 1 point, except for the combined exercise 17-24, which is worth 2 points.
  • Homework 5, due Tuesday, October 7. Same remarks as before apply. Section 7.1, exercises 20, 30, 32, 42. Section 7.2, exercise 40. Section 9.3, exercise 23. Each problem is worth 2 points; there are 2 extra credit points.
  • Homework 6, due Tuesday, October 14. Same remarks as before apply. Section 7.2, exercises 24, 32, 38, 42. Section 7.3, exercises 4, 11 32, 39. Section 7.4, exercises 4, 8, 12, 16, 26, 38, 46. Each exercise is worth 1 point.
  • Homework 7, due Tuesday, October 21. Same remarks as before apply. Section 7.4, exercise 50. Section 7.6, exercises 6, 10, 24, 27, 34. Practice Exercises for Chapter 7, exercises 46, 48, 98. Section 8.9, exercises 41, 42, 44. Each exercise is worth 1 point, and you may obtain two extra credit points.
  • Homework 8, due Tuesday, October 28. Same remarks as before apply. Section 7.7, exercises 11, 20, 32, 42, 50, 65, 74. Section 8.1, exercises 2, 8, 11, 12, 16, 42, 49, 81. Solve as many problems as you want; of those, up to 12 will be chosen randomly and graded. Each exercise is worth 1 point, so you may obtain two extra credit points.
  • Homework 9, due Tuesday, November 11. Same remarks as before apply. Solve at least 10 of the following problems; the others will be due as part of Homework 10. Do not use the solutions manual for any of these problems. Section 8.1, exercises 86, 88, 127. Also, the following exercise: Starting with a given x_0, define the subsequent terms of a sequence by setting x_{n+1}=x_n+\sin(x_n). Determine whether the sequence \{x_n\} converges, and if it does, find its limit. More precisely: You must indicate for which values of x_0 the sequence diverges, and for which it converges, and for those that converges, you must identify the limit, that may again depend on x_0. You may want to try studying the sequence with different initial values of x_0 (choose a large range of possible values) to get a feeling for what is going on. Section 8.2; exercises 14, 22, 38, 40 (do not use a calculator for this one; you can use that 2<e<3<\pi if necessary), 64-68, 71. Section 8.3; exercises 26, 35, 41, 43, 44.
  • Homework 10, due Tuesday, November 18. The usual considerations apply. Do not use the solutions manual for any of these problems. Turn in the problems listed for Homework 9  that you still have pending. Also: Section 8.4, exercises 3, 12, 23, 26, 35, 38, 40. Section 8.5, exercises 4, 8, 10, 25, 31, 34, 44, 47. Besides the exercises you have pending from last week, there are 15 new problems. Turn in the exercises you have pending, and at least 7 of the new problems. The others (at most 8) will be due December 2, together with the additional exercises for that week. Each exercise is worth 1 point.
  • Homework 11, due Tuesday, December 2, at the beginning of lecture. The usual considerations apply. Do not use the solutions manual for any of these problems. Turn in the problems listed for Homework 10  that you still have pending. Section 8.6, exercises 8, 25, 28, 37, 60. Section 8.7, exercises 2, 4, 39-48. Besides the exercises you have pending from last week, there are 17 new problems. Turn in the exercises you have pending, and at least 10 of the new problems. The others (at most 7) will be due December 9, together with the additional exercises for that week. Each exercise is worth 1 point.
  • Homework 12, due Tuesday, December 9, at the beginning of lecture. The usual considerations apply. Do not use the solutions manual for any of these problems. Turn in the problems you still have pending. Also: Section 8.8, exercise 28; section 8.9, exercise 40; section 8.10, exercise 19. 

Grading:

  • Homework: 60%.
  • Exam 1: 10%.
  • Exam 2: 10%.
  • Final exam: 20%.

I will then grade on a linear scale:

  • If your final score is 90% or higher, you receive an A.
  • If it is between 80 and 89%, you receive a B.
  • If it is between 70 and 79%, you receive a C.
  • If it is between 60 and 69% you receive a D.
  • If it is 59% or lower, you receive an F.
  • There may be a small curve up if the distribution of scores warrants this. Plus and minus grades might be used for grades near the top or bottom of a grade range.

Attendance: Not required, but encouraged. I will use this website to post any additional information, and encourage you to use the comments feature, but I will not post here content covered in class. If you leave a comment, please use your full name, which will simplify my life filtering spam out.

Core outcomes: In this class you will be assessed on a wide range of skills. Among these, the following make Math 175 a part of the University Core. By the end of the course, you should be able to:

  1. Identify and appropriately apply different integration techniques.
  2. Express solutions using (reasonably) correct mathematical language.
  3. Know that integration is an inverse operation to differentiation, and can be used to measure lengths, areas, and volumes, among others.
  4. Formally manipulate power series and justify rigorously these manipulations.
  5. Solve (separable) differential equations using the integration techniques covered throughout the course, and to express some of these solutions in terms of power series.

2 Comments | 175: Calculus II | Tagged: , | Permalink
Posted by andrescaicedo