1. barrels
Problem of the Day
200
100 6 12 18 24
hours
The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph above. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day?
A) 500 B) 600 C) 2, D) 3,000 E) 4,800

2. barrels
Problem of the Day
200
100 6 12 18 24
hours
The flow of oil, in barels per hour, through a pipeline on July 9 is given by the graph above. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day?
A) 500 B) 600 C) 2, D) 3,000 E) 4,800

3. Homework Questions?

4. 4-5: Integration By Substitution
Objectives:
Learn and practice the substitution technique for integration
Integrate even and odd functions
©2003 Roy L. Gover (

5. Review
Find the derivative:

6. Important Idea
The role of substitution in integration is comparable to the chain rule in differentiation.
Substitution, sometimes called u-substitution (u-sub), allows integration that otherwise could not be done.

7. Example
Find the
antiderivative: u
Then:
And:
Let:

8. Procedure Let u=the inside function. Find du in terms of dx.
Substitute and find the antiderivative in terms of u.
Substitute the expression for x in place of u.

9. Example
Find the antiderivative: u

11. Try This
Find the antiderivative using a u-substitution then differentiate your result:

13. Warm-Up
Solve the differential equation with the initial condition of y(0)=2:

15. Example
Find the antiderivative

17. Important Idea
The u-substitution must eliminate any product of variables.

19. Example
Find the antiderivative:

21. Example Find the antiderivative:
Ex 5, p295
This example requires a slightly different technique:

23. Try This
Find the antiderivative:
301/46

25. Example
Find the antiderivative:
Ex6,p296

27. Try This
Find the antiderivative:

29. Try This
Find the antiderivative:
Ex7a,p297

31. Example
Find the antiderivative:
Let u=?
Ex 7b,p297

32. Example
Find the antiderivative:
Ex7c,p 297

33. Try This
Find the antiderivative:
Ex7e,p 297

34. Assignment
304/7-25 odd & 31-37odd(find the general solution), odd Slides 1-21
2. 305/51-63 odd ,71,73 Slides 22-38

35. Warm-Up
Find the antiderivative:
Ex7d,p297

36. Important Idea
Substitution methods are also used with definite integrals. There are two methods:
Reverse the substitution and use original limits
Change the variable and use new limits

37. Example
Evaluate:

38. Try This
820

39. Example
Evaluate:

40. Definition
Odd functions are functions symmetric with the origin. They have exponents that are all odd.

41. Example
Odd Function

42. Definition
Even Functions are functions symmetric with the y axis. They have exponents that are all even. Constants are considered to have even exponents.

43. Example
Even Function

44. Try This
Even, odd or neither?

45. Important Idea
If f is an even function:

46. Important Idea
If f is an odd function:

47. Example
Evaluate:

48. Example
Evaluate:

49. Try This
Evaluate:

50. Lesson Close
Describe generally how you integrate a function by substitution.

51. Assignment
304/7-25 odd & 31-37odd(find the general solution), odd Slides 1-21
2. 305/51-63 odd ,71,73 Slides 22-38