TEST 1



Lecture note (last updated on Feb 11, 2007)

Dr. Firoz

Week 1-3 (Jan 16-Feb 2) Chapter 7 Section 7.1 and 7.8

Section 7.1 Integration by parts (Page # 475)

In this section we need the following formula for integration by parts:

[pic] , [pic] Fundamental theorem of calculus.

Formulas to remember for this chapter:

[pic]

Examples.

1. [pic], when [pic]

[pic] and [pic]

2.[pic], when [pic], [pic]

[pic] and also we consider [pic]

3.[pic], when [pic]

[pic] and [pic]also

[pic]= [pic]+ C consider [pic]

4.[pic], when [pic]

[pic] and [pic]also

5. [pic], when [pic]

[pic] and [pic]also

[pic]

6. [pic]. We consider [pic] and [pic]and also [pic].

We have now [pic]. Now try to use integration by parts to find that [pic]

[pic]

OR: We consider [pic] and [pic]

Then we have [pic]

Now [pic]

7. [pic] [pic]

[pic] [pic]

8. [pic] [pic]

[pic] [pic]

[pic]

9. [pic] [pic]

[pic] [pic]

[pic]

10. Prove the reduction formula [pic]

Let us consider [pic] and [pic]

Now

[pic]

11. Use the reduction formula in example 9 to prove that

[pic]

We have [pic]

Plug [pic] for n: to get

[pic] … … (1)

[pic]… …. (2)

Now let us plug [pic], [pic], [pic], …3, 2 successively for 2n in (2): to get the following results:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Finally from (2) we have the following result:

[pic]

Homework Problems Page # 480:

6. [pic]

where [pic]

7.

[pic]

where [pic]

12. See example 5 above.

20. [pic]

21. See example 6 above.

34. [pic] compare with the problem we have solved in the class. Look at your note.

44. We have solved in the class.

46. [pic]

where [pic]

Section 7.2 Trigonometric Integrals (Page # 482)

Following trigonometric identities are useful:

[pic]

[pic]

Example 1 Evaluate [pic]

Or we could solve by substitution as

[pic]

where [pic]

Thus we have [pic].

Our results are equivalent.

Example 2 Evaluate [pic]

Where [pic] and [pic]

Example 3 Evaluate [pic]

[pic]

We have

[pic], where [pic]. Do the rest!

Problem 40. [pic]

[pic]

where

[pic]

Problem 51 [pic] . Use formula [pic]

Thus we have [pic]

Notice that [pic] whenever F has a horizontal tangent. Look at the graphs below. We have[pic] (red graph) and [pic] (green graph) when C = 0.

➢ plot([sin(3*x)*sin(6*x),1/6*sin(3*x)-1/18*sin(9*x)], x=-3..3, thickness=2);

Solution graph from question number 51.

[pic]

Learn more on plot in Maple using > ?plot

Homework Problems Page # 488:

6. [pic]

where [pic]

8. [pic]

[pic], because

where [pic]

Section 7.3 Trigonometric Substitution (Page# 489)

Homework Hints: You need to complete.

5. [pic]

[pic], where [pic]

[pic]

6. [pic]

[pic], where [pic]

Next you plug [pic]to compete the integral

8. [pic]

[pic], where [pic]

14. [pic]

[pic], where [pic]

21. [pic]

[pic]

where [pic]

OR: [pic], [pic]

26. [pic]

[pic]

Next put [pic]

32. [pic]

a) put [pic]

[pic]

b) put [pic]

[pic], now substitute back what is t.

Section 7.4 Integration of rational functions by partial fraction (Page# 496)

Homework Hints: You need to complete.

4. a) [pic]

b) [pic]

14. [pic]

[pic]

[pic]

18. [pic], find that [pic]

19. [pic], find that [pic]

22. [pic], find that [pic]

26. [pic], find that [pic]

28. [pic], find that [pic]

44. Put[pic]and use the formula [pic]

50. [pic], proceed with integration by parts [pic]

54. [pic]

which is performed by first completing square and then substitution method for the first part and apply formula (6) for the second part.

This problem also could be solved (if method is not mentioned):

[pic], and then use partial fraction.

Section 7.5 Strategy for Integration (Page# 505)

Homework Hints: You need to complete.

4. [pic], where [pic]

Now apply formula [pic]

6. [pic]

Precede using integration by parts [pic]

7. [pic]

Proceed using integration by parts [pic]. Do not forget to find definite integral result.

8. [pic], Factor the denominator and apply integration by partial fraction.

14. [pic], proceed by substitution [pic] .

15. [pic], Factor the denominator and apply integration by partial fraction.

16. [pic], substitute [pic]

25. [pic] by long division. Now use partial fraction.

33. [pic], use substitution [pic]

34. [pic], substitution [pic]

Section 7.6 Integration Using Tables and computer Algebra Systems (Page# 511)

Homework Hints: You need to complete.

Tables of indefinite integrals are very useful when we are confronted by an integral that is difficult by hand or very lengthy in calculation and we do not have access to a computer algebra system. We have a list of few such integrals on the reference pages at the end of our text book. Those formulas we do not need to memorize, rather those will be provided on the test. In this section we will learn how to use those given formulas.

Example 1. (13) [pic]

This problem we could solve without using formula 69. Just substitute [pic]

[pic], look at example 7, page number 487.

Example 2. (6) [pic], substitute [pic]

[pic], 45 above equal sign means we are using formula 45 from our text book. You need to evaluate the result at the given limits.

Example 3. (16)[pic], substitute [pic]

[pic], using the trigonometric formula [pic]. Or it could be solved just using formula 81.

Example 4. (22) [pic], substitute [pic]

[pic]

Example 5. (36) [pic], but if you do it in Maple you may have result like [pic]. The results are equivalent one can check using binomial expansion.

Example 6. (38) [pic], by Maple

By hand one can substitute [pic], then

[pic]. The results are equivalent if you substitute [pic]

43. When we type [pic]CAS recognize it as a promising substitute because we have another tern outside of square root sign.

44. We need to substitute [pic]

Section 7.7 Approximate Integration (Page# 518)

Homework Hints: You need to complete.

There are two situations in which we can not find exact value of a definite integral or it is impossible to evaluate. The fisr situation arises from the fact that in order to evaluate the definite integral [pic]using the fundamental theorem of calculus we do not know the antiderivative of f(x). We have seen some examples in section 7.5, where our integrals are rather difficult. Also we have some cases where the integrals are impossible like [pic]. Second we may have some practical problem which does not fit with any formulas or simplifications. In this section we will use only three methods.

Mid Point Rule: [pic], where step size =[pic] and [pic], also [pic]

The error estimate is [pic]where [pic]for [pic]

Trapezoidal Rule: [pic], where

step size =[pic] , [pic]

The error estimate is [pic]where [pic]for [pic]

Simpson’s Rule: [pic], where

step size =[pic] , [pic]

The error estimate is [pic]where [pic]for [pic]

Note: [pic]

Example 1. Approximate the integral [pic] for [pic] using

a) Mid point rule ([pic]) b) Trapezoidal rule ([pic]) and c) Simpson’s rule (n = 10) d) Find error estimations in each rule. e) Verify that [pic]

a) Mid point rule [pic], [pic]

[pic]

[pic]

b) Trapezoidal rule [pic]

[pic]

c) Simpson’s rule [pic]

[pic]

d) Error bounds for [pic]; [pic]

[pic]where [pic]for [pic]

[pic]

[pic] where [pic]for [pic]

e) [pic]which is true.

Example 2. How large should we take n in order to guarantee that the trapezoidal rule midpoint rule approximations of [pic] are accurate to within 0.0001?

From example 1 we have k = 2. [pic]and [pic]

[pic] (because n must be even)

Answers to Home work 1 problems (Even numbers only)

Section 7.1

6. [pic] 12. [pic]

20. [pic] 34. [pic]

Section 7.2

6. [pic] 8. [pic] 14. [pic] 24. [pic]

30. [pic] 40. [pic]

Section 7.3

6. [pic] 8. [pic] 14. [pic] 26. [pic]

32. a) [pic] b) [pic]

Section 7.4

4. a) [pic]

b)[pic]

14. [pic] , when [pic]; [pic], when a = b

18. [pic] 22. [pic]

26. [pic] 28. [pic]

44. [pic] 50. [pic]

54. [pic]

Section 7.5

4. [pic] 6. [pic] 8. [pic] 26.

14. [pic] 16. [pic] 30. [pic] 34. [pic]

For Test 1 you need to remember the following formulas:

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic]

[pic]

-----------------------

MAT 271

SPRING 2007

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download