1. Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

2. Area How do we calculate the area of some irregular shape? For example, how do we calculate the area under the graph of f on [a,b]?

3. Area Approach: number of rectangles: width of each rectangle: We approximate the area using rectangles.

4. Area Left-hand estimate: Let the height of each rectangle be given by the value of the function at the left endpoint of the interval.

5. Area Left-hand estimate: Riemann Sum

6. Area Right-hand estimate: Let the height of each rectangle be given by the value of the function at the right endpoint of the interval.

7. Area Right-hand estimate: Riemann Sum

8. Area Midpoint estimate: Let the height of each rectangle be given by the value of the function at the midpoint of the interval.

9. Area Midpoint estimate: Riemann Sum

10. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

11. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

12. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

13. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact?

14. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

15. Riemann Sums and the Definite Integral Definition: The definite integral of a function on the interval from a to b is defined as a limit of the Riemann sum where is some sample point in the interval and

16. The Definite Integral Interpretation: If, then the definite integral is the area under the curve from a to b.

17. Example Estimate the following definite integrals using left-endpoints, midpoints, and right-endpoints and the indicated number of intervals. (a) (b)

18. Estimating Using Left-Endpoints

20. Estimating Using Right-Endpoints

22. Estimating Using Midpoints

24. Estimating Using Left-Endpoints

26. Estimating Using Right-Endpoints

28. Estimating Using Midpoints

30. Types of Integrals Indefinite Integral Definite Integral antiderivative of f function of x number

31. The Definite Integral Interpretation: If is both positive and negative, then the definite integral represents the NET or SIGNED area, i.e. the area above the x-axis and below the graph of f minus the area below the x-axis and above f

32. Definite Integrals and Area Example: Evaluate the following integrals by interpreting each in terms of area. (a)(b) (c)

33. Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.

34. Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.

35. Summation Property of the Definite Integral (6) Suppose f(x) is continuous on the interval from a to b and that Then

36. Properties of the Definite Integral (7) Suppose f(x) is continuous on the interval from a to b and that Then

37. Area How do we calculate the area of some irregular shape? For example, how do we calculate the area under the graph of f on [a,b]?

38. Area Approach: number of rectangles: width of each rectangle: We approximate the area using rectangles.

39. Area Left-hand estimate: Let the height of each rectangle be given by the value of the function at the left endpoint of the interval.

40. Area Left-hand estimate: Riemann Sum

41. Area Right-hand estimate: Let the height of each rectangle be given by the value of the function at the right endpoint of the interval.

42. Area Right-hand estimate: Riemann Sum

43. Area Midpoint estimate: Let the height of each rectangle be given by the value of the function at the midpoint of the interval.

44. Area Midpoint estimate: Riemann Sum

45. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

46. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

47. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

48. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact?

49. Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

50. Riemann Sums and the Definite Integral Definition: The definite integral of a function on the interval from a to b is defined as a limit of the Riemann sum where is some sample point in the interval and

51. The Definite Integral Interpretation: If, then the definite integral is the area under the curve from a to b.

52. Example Estimate the following definite integrals using left-endpoints, midpoints, and right-endpoints and the indicated number of intervals. (a) (b)

53. Estimating Using Left-Endpoints

55. Estimating Using Right-Endpoints

57. Estimating Using Midpoints

59. Estimating Using Left-Endpoints

61. Estimating Using Right-Endpoints

63. Estimating Using Midpoints

65. Types of Integrals Indefinite Integral Definite Integral antiderivative of f function of x number

66. The Definite Integral Interpretation: If is both positive and negative, then the definite integral represents the NET or SIGNED area, i.e. the area above the x-axis and below the graph of f minus the area below the x-axis and above f

67. Definite Integrals and Area Example: Evaluate the following integrals by interpreting each in terms of area. (a)(b) (c)

68. Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.

69. Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.

70. Summation Property of the Definite Integral (6) Suppose f(x) is continuous on the interval from a to b and that Then

71. Properties of the Definite Integral (7) Suppose f(x) is continuous on the interval from a to b and that Then

72. Announcements Topics: -sections 7.3 (definite integrals), 7.4 (FTC), and 7.5 (additional techniques of integration) * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

73. Types of Integrals Indefinite Integral Definite Integral antiderivative of f function of x number

74. The Fundamental Theorem of Calculus If is continuous on then where is any antiderivative of, i.e.,

75. Evaluating Definite Integrals Example: Evaluate each definite integral using the FTC. (a)(b) (c)(d)

76. Evaluating Definite Integrals Example: Try to evaluate the following definite integral using the FTC. What is the problem?

77. Differentiation and Integration as Inverse Processes If f is integrated and then differentiated, we arrive back at the original function f. If F is differentiated and then integrated, we arrive back at the original function F. FTC II FTC I

78. The Definite Integral - Total Change Interpretation: The definite integral represents the total amount of change during some period of time. Total change in F between times a and b: value at end value at start rate of change

79. Application – Total Change Example: Suppose that the growth rate of a fish is given by the differential equation where t is measured in years and L is measured in centimetres and the fish was 0.0 cm at age t=0 (time measured from fertilization).

80. Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?

81. Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?

82. Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?