Math 142

Chapter 8 Homework

Homework will be posted as it becomes available.

section

handouts

note to yourself
of due date

homework problems
in case the bookstore runs out of books    Calculus is so popular that sometimes the bookstore runs out of Calculus books at the beginning of the semester and has to order more. If you are currently without of book because of this, below are copies of the first few sections of the course (along with the answer to the odds in the back of the book). Of course, you must have a book for the semester so you will need to get one. The below posting is just to tie you over until more books arrives to the bookstore, at which point copies of the sections will not be posted. When we finish the lectures for the below sections, please let me know if the books are still not at the bookstore and I will further investigate the problem.
Sections: 8.1  &  8.2  &  8.3
8.0   Do as much as you need to do from the preparation for Math 142
Integration Basics   This Integration Basics handout collects some basics from Math 141 that you need to know in Math 142. I highly suggest you take a look the first problem from any Exam 1 since 2006 from my previous exams. On our Exam 1:
If you do not make at least half of the points on Problem 0, then your score for the entire exam will be whatever you made on Problem 0. There really is no need to look further through your exam.
A practice quiz to help reinforce the Integration Basics handout.
8.1

Integration by u-du Substitution

Calc I material

 §8.1 covers u-du substitution, which is Calc I material and thus no lecture will be given on §8.1. Below is a u-du sub. review, which you should do.
First we review u-du sub. from another book.
  • Work through Examples 1-11 from here (another book's §5.5 u-du sub.)
  • From this other book's 88 exercises, do: 13, 17, 19, 25, 33, 43, 59, 67, 81, 88 (hint: cos(x) = sin (π/2 - x) and so ∫ 0π/2 f(cos(x)) dx = ∫ 0π/2 f(sin (π/2 - x)) dx and to integrate the latter let u = π/2 - x).
  • Answers to the odd problems. Give the exercises an honest shot before peeking at the solutions to all of them.
    • Stewart (6th ed, ET), §5.5.
    Next, back to our book (by Thomas).
  • Review §5.5 and 5.6 (p 339-356) on u-du sub., which you covered in Calc I.
  • Work through § 8.1's Examples 1-8 (pg 456-460).
  • From § 8.1, do: 5 (hint: write the integral as a sum of 2 integrals - this type of integral will appear often in §8.4Trig Subst.), 9 (hint: use algebra to simplify the integrand), 11 (watch your limits of integration when you do a u-du subst.), 21 (hint: see Example 5 on p. 458 - we will explore integrals of this form further in §8.5Partial Fractions), 41, 43 (see §6.3, formula (3), p385).
    • need more practice? Having troubles or want more practice? Work through some more of the above 88 exercises. This is a good source and you have the solutions.
    need more practice? Want or need even more practice with u-du substitution?
    Try these 23(=30-7) u-du substitution problem (answers included).
    Anton (8th ed, ET)
    8.2

    Parts

    		   9, 11, 13, 23, 25, 27, 29, 53, 65.

    In lecture on 1/12/17, we got through 3 of the 5 Key Ideas from Parts, at which point you should be able to do all of the above problems except: 27, 65. Recitation on Wed (1/18) will be over parts so it is best to get started on parts (8.2).

    Parts is just one (of many) techniques (of integration) we will learn in this chapter. Which technique to use on a given integral - a good ansatz is a techique already taught. Thus far, in this second section in the chapter on Techniques of Integration, we have only 2 techniques in our tool box: substitution and parts. As we learn more techniques, it becomes tricker to answer the question: which technique to use? The answer lies in true understanding and pattern recognition (useful skills). So take out a sheet of paper and do the following.

    • Recall the 5 KEY IDEAS from Parts taught in class (also on the Integration Basics handout). Write out each Key Idea, leaving space between the key ideas.
    • Under each Key Idea, list out the specific integrals from the homework assignment that used that Key Idea. (e.g., Under "Bring to the other side method", write "23. ∫ e2x cos 3x dx").
    • Go back and look for patterns/hints on why and how parts works.
    • Did this help? If not - do more problems.
    need more practice? Try these exercises. Answers to the odd problems. Give the exercises an honest shot before peeking at the solutions to all of them.
    Stewart (6th ed, ET)
    You do not need to know tabular integration, which basically is a method (which students tend to misuse) to accomplish something that can easily be done with high-school algebra.
    8.1&82.     A Sample Recitation Quiz over Sections 8.1&8.2, along with the solution. Would you be able to work this quiz at the end of a recitation over Sections 8.1&8.2, in 10 minutes, without your books/notes?
    8.3

    trig integrals

    		 handout   7, 13, 15, 17, 23, 25, 31, 35, 37, 43, 49, 53, 55, 59, 69 (see §6.3, formula (3), p385).
    need more practice? Try these exercises. Answers to the odd problems. Give the exercises an honest shot before peeking at the solutions to all of them.
    Stewart (6th ed, ET)
    8.4

    trig substitution

    		 handout   1, 7, 11, 15, 23, 35, 37, 47, 54 (ans: Πab), 55a, 57. Also do these (click here) two problems.
     need more practice? Try these exercises. Answers to the odd problems. Give the exercises an honest shot before peeking at the solutions to all of them.
    Stewart (6th ed, ET)
    8.5

    partial fractions

    		handout   General hint (esp. helpful for the definite integrals): thanks to the FTC, you can check your answer to an indefinite integral by making sure the derivative of your answer is the integrand.
    3, 5, 13, 17, 21, 27, 35, 37, 39, 43, 45 .
     need more practice? Try these exercises. Answers to the odd problems. Give the exercises an honest shot before peeking at the solutions to all of them.
    Stewart (6th ed, ET)
    Review of
    8.1-8.5    		   Time to test your pattern recognition skills (valuable skill no matter where your academic pursuit leads you). For each of Sections 8.1 through 8.5, look over the section's examples with an eye on what properties are shared by integrals that can be solved using that section's technique.
    100 Integrals Handout   Next start on the 100 Integrals handout, which has 100 integrals along with 19 applications of integration.
    1. You should work 10 integrals a day, 7 days a week, until you finish all these integrals.
    2. Omit numbers: 17, 51, 72, 74, 96, 97, 101-105, 112, 113.
    3. The answers to the odd number problems.
    4. Prof. Girardi's handwritten solutions to the circled problems: 1, 3, 7, 23, 26, 35, 36, 47, 50, 71.
    5. 28 pages of hints/answers/solutionsfor all (#1-119) the problems.
    Edwards&Penny (3rd ed)
     need more practice? Try these 81 integrals. Answers to the odd problems. Give the exercises an honest shot before peeking at the solutions to all of them.
    Stewart (6th ed, ET)
    8.6 We are not covering this section on Tables and CAS's.
    8.7
    Numerical Intergation    		 handout For the book's problem 3, answer questions 1-4 from this handout. (answers: (1) 11⁄4, (2) 2, (3) 1⁄12, (4) 116.)
    pre-8.8Indeterminate Forms & L'Hopital's Rule (from Calc I) are used in §8.8. If you to review, see "Preparation for Math 142" on the homework page.
    8.8
    Improper Integrals    		 For § 8.8, use the below source (not the textbook) for homework.
    1. homework problems
    2. answers to odd
    3. solutions to all (use responsiblity)
    Remarks on the above solutions.
    • The solutions often refer to Equation 2. This is our book's Example 3 on page 507, which is
      ∫1∞ 1⁄xp dx     is convergent if p>1 and divergent if p ≦ 1.
      We did this example in class.
    • There are different ways on integral can be improper. Be able to explain, in an English sentence and not by a number (1,2,3,4,5,6,I,orII), why an integral is improper. For the purpose of learning, I divided the ways into numbers 1-6 on the Improper Integrals handout in class. Just as our book does (see the definitions on pages 505&508), this homework source calls ways 1-3 of Type I and ways 4-6 of Type II.
    Improper Integrals without comparison tests (DCT/LTC)

    2017 Spring: did before Exam 1.

    • To see if you have the basic ideas down, do this Practice Problem 0 for Improper Integrals (soln)
    • For assigned problems within 5-40, for a divergent improper integral, explain why it diverges (eg., dvg to ∞, dvg to -∞, dvg but not to ±∞).
    • From the above (not the textbook) source of problems:
      1, 2, 5, 7(dvg to ∞), 13, 15(dvg but not to ±∞), 19, 23, 27(dvg to ∞) , 31(dvg to ∞), 41, 59, 61, 71.
    DCT/LCT for Improper Integrals with comparison tests (DCT/LCT)

    2017 Spring: Question and Quiz on Wed 2/22.

    		For the above (not the textbook) source of problems, use the specified comparison test (DCT or LCT) to determine if the integral is conervgent or divergent.
    • use DCT for 49, 50, 51, 52.
    • use LCT for the following slight variants of 49, 50, 51. The variants are chosen so that one compares the integrand to the same g as in the original problem but, in the variant, the DCT does not easily work (why? check which way the bound easily goes) but the LCT easily works. Remark/Hint.
      49 but change (x)⁄(x3 + 1) to (x)⁄(x3 ‒ 1) . Ans: convergent.
      50 but change (2 + e-x)⁄(x) to (2 ‒ e-x)⁄(x). Ans: divergent.
      51 but change (x+1)⁄ √x4 ‒x to (x+1)⁄ √x4 +x. Ans: divergent.
    Laplace transform

    2017 Spring: Question and Quiz on Wed 2/22.

    		For the above (not the textbook) source of problems: 71, 72, 73
    (an introducation to the Laplace transform, which is used in many branches of the sciences and engineering)
    Then Exam 1 over 8.1-8.5, 8.7, 8.8.

    Findable from URL: http://people.math.sc.edu/girardi/w142.html