1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 5 The Definite Integral

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.1 Estimating with Finite Sums

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 5 Quick Review Solutions

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 6 What you’ll learn about Distance Traveled Rectangular Approximation Method (RAM) Volume of a Sphere Cardiac Output … and why Learning about estimating with finite sums sets the foundation for understanding integral calculus.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 7 Example Finding Distance Traveled when Velocity Varies

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 8 Example Finding Distance Traveled when Velocity Varies

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 9 LRAM, MRAM, and RRAM approximations to the area under the graph of y=x 2 from x=0 to x=3

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 10 Example Estimating Area Under the Graph of a Nonnegative Function

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.2 Definite Integrals

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 12 Quick Review

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 13 Quick Review Solutions

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 14 What you’ll learn about Riemann Sums The Definite Integral Computing Definite Integrals on a Calculator Integrability … and why The definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 15 Sigma Notation

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 16 The Definite Integral as a Limit of Riemann Sums

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 17 The Existence of Definite Integrals

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 18 The Definite Integral of a Continuous Function on [a,b]

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 19 The Definite Integral

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 20 Example Using the Notation

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 21 Area Under a Curve (as a Definite Integral)

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 22 Area

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 23 The Integral of a Constant

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 24 Example Using NINT

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.3 Definite Integrals and Antiderivatives

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 26 Quick Review

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 27 Quick Review Solutions

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 28 What you’ll learn about Properties of Definite Integrals Average Value of a Function Mean Value Theorem for Definite Integrals Connecting Differential and Integral Calculus … and why Working with the properties of definite integrals helps us to understand better the definite integral. Connecting derivatives and definite integrals sets the stage for the Fundamental Theorem of Calculus.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 29 Rules for Definite Integrals

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 30 Example Using the Rules for Definite Integrals

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 31 Example Using the Rules for Definite Integrals

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 32 Example Using the Rules for Definite Integrals

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 33 Average (Mean) Value

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 34 Example Applying the Definition

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 35 The Mean Value Theorem for Definite Integrals

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 36 The Mean Value Theorem for Definite Integrals

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 37 The Derivative of an Integral

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 38 Quick Quiz Sections 5.1 - 5.3

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 39 Quick Quiz Sections 5.1 - 5.3

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 40 Quick Quiz Sections 5.1 - 5.3

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 41 Quick Quiz Sections 5.1 - 5.3

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 42 Quick Quiz Sections 5.1 - 5.3

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 43 Quick Quiz Sections 5.1 - 5.3

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.4 Fundamental Theorem of Calculus

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 45 Quick Review

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 46 Quick Review Solutions

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 47 What you’ll learn about Fundamental Theorem, Part 1 Graphing the Function Fundamental Theorem, Part 2 Area Connection Analyzing Antiderivatives Graphically … and why The Fundamental Theorem of Calculus is a Triumph of Mathematical Discovery and the key to solving many problems.

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 48 The Fundamental Theorem of Calculus

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 49 The Fundamental Theorem of Calculus

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 50 Example Applying the Fundamental Theorem

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 51 Example The Fundamental Theorem with the Chain Rule

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 52 Example Variable Lower Limits of Integration

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 53 The Fundamental Theorem of Calculus, Part 2

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 54 The Fundamental Theorem of Calculus, Part 2

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 55 Example Evaluating an Integral

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 56 How to Find Total Area Analytically

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 57 How to Find Total Area Numerically

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.5 Trapezoidal Rule

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 59 Quick Review

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 60 Quick Review Solutions

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 61 What you’ll learn about Trapezoidal Approximations Other Algorithms Error Analysis … and why Some definite integrals are best found by numerical approximations, and rectangles are not always the most efficient figures to use.

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 62 Trapezoidal Approximations

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 63 The Trapezoidal Rule

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 64 Simpson’s Rule

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 65 Error Bounds

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 66 Quick Quiz Sections 5.4 and 5.5 x1467 f(x)10304020

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 67 Quick Quiz Sections 5.4 and 5.5 x1467 f(x)10304020

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 68 Quick Quiz Sections 5.4 and 5.5

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 69 Quick Quiz Sections 5.4 and 5.5

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 70 Quick Quiz Sections 5.4 and 5.5

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 71 Quick Quiz Sections 5.4 and 5.5

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 72 Chapter Test

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 73 Chapter Test

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 74 Chapter Test 8. A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. (a)Give an upper estimate and a lower estimate for the amount of oil consumed by the generator during that week. (b)Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week.

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 75 Chapter Test

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 76 Chapter Test Solutions

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 77 Chapter Test Solutions 3 4 Let R be the region in the first quadrant enclosed by the x-axis and the graph of the function y  4x – x. 1. Sketch the rectangles and compute byhand the area for the MRAM approximations 4.125. 2. Sketch 4 the trapeziods and compute by hand the area for the T approximations. 3.75

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 78 Chapter Test Solutions

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 79 Chapter Test Solutions 8. A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. (a)Give an upper estimate and a lower estimate for the amount of oil consumed by the generator during that week. Upper = 4.392 L; Lower = 4.008 L (b)Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week. 4.2L

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 80 Chapter Test Solutions