Math 142

Chapter 8 Homework

Homework will be posted as it becomes available.

section

handouts

remark

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

Basic Skills Check

   Online HW
For sources of help with this online homework set, see
  1. optional preparation homework at preparation for Math 142
  2. the course handouts listed under Preparation at Handouts.
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

 Online HW
§8.1 covers u-du substitution, which is Calc I material and thus no lecture will be given on §8.1. In our calculus book (by Thomas), u-du sub was covered in §5.5&5.6. For lots of worked out examples of u-du substitution, see §6.1 (pages 263-282) of the APEX Calculus book (a great book for example!).
Hints for Online HW (2nd number refers to problem from book in §8.1)
  1. =8.1.9. Note that e-x = 1⁄ ex. So the denumerator of the fraction
    (1) / [e-x + ex ] is really a fraction itself. So we algebraically clean up this fraction divided by a fraction as:
    (1) / [e-x + ex ] = (1) / [ (1 / ex)+ ex ] = (1) / [ (1 / ex)+ (ex ex / ex )] = (1) / [ (1+ e2x) / (ex)] = (ex) / [1+ e2x].
    Hopefully this helps motivate a MML-hint that to simplify the expression such as (1)⁄[e-x + ex], one just multiplies the expression by (ex)⁄[ex], which is just the number 1, cleaverly written.
  2. =8.1.11. Do a u-du sub. to express the integral as an integral from page 2 of your blue sheet. Do not forget that, when doing definite integral, if you have a dx then your limits of integration must be as x varies (so from x= to x= ) while if you have a du then your limits of integration must be as u varies.
  3. =8.1.36. Complete the square under the square root sign and then do a u-du sub. to express the integral as an integral from page 2 of your blue sheet.
  4. =8.1.3. Preview of to-come §8.3 (Trig Integrals). Using basic trig identities, rewrite the integral as a sum of integrals with each one do-able, perhaps after a u-du substitutions, using page 2 of your blue sheet.
  5. =8.1.5. Preview of to-come §8.4 (Trig Sub). Rewrite the integral as a sum of two integrals where each integral, after the proper u-du substitutions, is do-able using page 2 of your blue sheet.
  6. =8.1.21. Preview of to-come §8.5 (Partial Fractions). The integrand is a rational function, i.e., a polynomial y=p(t) divided by another polyomial y=q(t). We will learn that if you want to integrate such a rational function [p(t)]⁄[q(t)] and we do NOT have "strictly bigger bottoms" (i.e., the degree of q is NOT stricly bigger than the degree of p), then first rewrite the rational function by doing long division of polynomial. In the given problem, since the degree of the bottom q is 2 while the degree of the top p is 3, we do NOT have "strictly bigger bottoms". So do long division to rewrite the given rational function as a sum of fucntions, each of whose integral, after the proper u-du substitutions, is do-able using page 2 of your blue sheet.
  7. =8.1.41. Express the area as an integral (see §5.6).
  8. =8.1.43. Express the arc length as an integral (see §6.3).
Below is a u-du sub. review from another book.
  • Read (and 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.
    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

    		   Online HW
    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.
    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)
    8.1&8.2     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

    		   Online HW
    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

    		   Online HW
    Also do these (click here) two problems (hint: complete the square)
     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

    		  Online HW
     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.
    81 Integrals  Do these 81 Integrals
    1. You should work 10 integrals a day, 7 days a week,
      until you finish all these integrals.
    2. Answers to the odd problems.
    3. Give the exercises an honest shot before peeking at the solutions to all of them.

    Stewart (6th ed, ET)
    100 Integrals need more practice? Try these 100 Integrals
    1. Omit numbers: 17, 51, 72, 74, 96, 97, 101-105, 112, 113.
    2. The answers to the odd number problems.
    3. Prof. Girardi's handwritten solutions to the circled problems: 1, 3, 7, 23, 26, 35, 36, 47, 50, 71.
    4. 28 pages of hints/answers/solutionsfor all (#1-119) the problems.
    Edwards&Penny (3rd ed)
    8.6 We are not covering this section on Tables and CAS's.
    8.7No Online HW. HW is as given in a Maple Lab.
    8.8
    Improper Integrals

     

     

     

    without comparison tests (DCT/LTC)

     

     

     

    and

     

     

     

    with comparison tests (DCT/LTC)

    		For §8.8 OFF-LINE homework, use the below source (note, not our textbook).
    1. homework problems
    2. answers to odd
    3. solutions to all (use responsiblity)
    For problems within 5-40, for a divergent improper integral, explain why it diverges (eg., dvg to ∞, dvg to -∞, dvg but not to ±∞).

    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.
    ON-LINE HWTwo Online HW sets.
    One set without, and one set with, the DCT/LCT.
    Take notice of the number of attempts per problem for each set.
    OFF-LINE HW

    with comparison tests (DCT/LCT)

    		 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.
    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)
    need more practice? For more practice without the comparison tests, from the above (not the textbook) source of problems, a good selection is:
    1, 2, 5, 7(dvg to ∞), 13, 15(dvg but not to ±∞), 19, 23, 27(dvg to ∞) , 31(dvg to ∞), 41, 59, 61, 71.
    Then Exam 1 over 8.1-8.5, 8.8.

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