1. ESSENTIAL CALCULUS CH04 Integrals

2. In this Chapter: 4.1 Areas and Distances 4.2 The Definite Integral 4.3 Evaluating Definite Integrals 4.4 The Fundamental Theorem of Calculus 4.5 The Substitution Rule Review

3. Chapter 4, 4.1, P194

4. Chapter 4, 4.1, P195

9. Chapter 4, 4.1, P196

10. Chapter 4, 4.1, P197

12. Chapter 4, 4.1, P198

14. Chapter 4, 4.1, P199

15. 2. DEFINITION The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: A=lim R n =lim[f(x 1 )∆x+f(x 2 ) ∆ x+ ‧‧‧ +f(x n ) ∆ x] n → ∞

16. Chapter 4, 4.1, P199

17. This tells us to end with i=n. This tells us to add. This tells us to start with i=m.

18. Chapter 4, 4.1, P199

19. Chapter 4, 4.1, P200 The area of A of the region S under the graphs of the continuous function f is A=lim[f(x 1 )∆x+f(x 2 ) ∆x+ ‧‧‧ +f(x n ) ∆x] A=lim[f(x 0 )∆x+f(x 1 ) ∆x+ ‧‧‧ +f(x n-1 ) ∆x] A=lim[f(x* 1 )∆x+f(x* 2 ) ∆x+ ‧‧‧ +f(x* n ) ∆x] n → ∞

20. Chapter 4, 4.2, P205 FIGURE 1 A partition of [a,b] with sample points

21. Chapter 4, 4.2, P205 A Riemann sum associated with a partition P and a function f is constructed by evaluating f at the sample points, multiplying by the lengths of the corresponding subintervals, and adding:

22. Chapter 4, 4.2, P206 FIGURE 2 A Riemann sum is the sum of the areas of the rectangles above the x-axis and the negatives of the areas of the rectangles below the x-axis.

23. 2. DEFINITION OF A DEFINITE INTEGRAL If f is a function defined on [a,b],the definite integral of f from a to b is the number provided that this limit exists. If it does exist, we say that f is integrable on [a,b]. Chapter 4, 4.2, P206

24. NOTE 1 The symbol ∫was introduced by Leibniz and is called an integral sign. It is an elongated S and was chosen because an integral is a limit of sums. In the notation is called the integrand and a and b are called the limits of integration; a is the lower limit and b is the upper limit. The symbol dx has no official meaning by itself; is all one symbol. The procedure of calculating an integral is called integration.

25. Chapter 4, 4.2, P206

26. Chapter 4, 4.2, P207 3. THEOREM If f is continuous on [a,b], or if f has only a finite number of jump discontinuities, then f is integrable on [a,b]; that is, the definite integral dx exists.

27. Chapter 4, 4.2, P207 4. THEOREM If f is integrable on [a,b], then where

28. Chapter 4, 4.2, P208

32. Chapter 4, 4.2, P210

33. Chapter 4, 4.2, P211

34. MIDPOINT RULE where and

35. Chapter 4, 4.2, P212

37. Chapter 4, 4.2, P213

39. PROPERTIES OF THE INTEGRAL Suppose all the following integrals exist. where c is any constant

40. Chapter 4, 4.2, P214

42. COMPARISON PROPERTIES OF THE INTEGRAL 6. If f(x)≥0 fpr a≤x≤b. then 7.If f(x) ≥g(x) for a≤x≤b, then 8.If m ≤f(x) ≤M for a≤x≤b, then

43. Chapter 4, 4.2, P215

44. Chapter 4, 4.3, P217 29.The graph of f is shown. Evaluate each integral by interpreting it in terms of areas. (a) (b) (c) (d)

45. Chapter 4, 4.3, P217 30. The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral. (a) (b) (c)

46. Chapter 4, 4.3, P218 EVALUATION THEOREM If f is continuous on the interval [a,b], then Where F is any antiderivative of f, that is, F ’ =f.

47. Chapter 4, 4.3, P220 the notation ∫f(x)dx is traditionally used for an antiderivative of f and is called an indefinite integral. Thus The connection between them is given by the Evaluation Theorem: If f is continuous on [a,b], then

48. Chapter 4, 4.3, P220 ▓ You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions).

49. Chapter 4, 4.3, P220 1. TABLE OF INDEFINITE INTEGRALS

50. Chapter 4, 4.3, P221 ■ Figure 3 shows the graph of the integrand in Example 5. We know from Section 4.2 that the value of the integral can be interpreted as the sum of the areas labeled with a plus sign minus the area labeled with a minus sign.

51. Chapter 4, 4.3, P222 NET CHANGE THEOREM The integral of a rate of change is the net change:

52. Chapter 4, 4.4, P227 The Fundamental Theorem deals with functions defined by an equation of the from

53. Chapter 4, 4.4, P227

57. Chapter 4, 4.4, P229

58. THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1 If f is continuous on [a,b], then the function defined by a≤x≤b is an antiderivative of f, that is, g ’ (x)=f(x) for a<x<b.

59. Chapter 4, 4.4, P231 THE FUNDAMENTAL THEOREM OF CALCULUS Suppose f is continuous on [a,b]. 1.If g(x)= f(t)dt, then g ’ (x)=f(x). 2. f(x)dx=F(b)-F(a), where F is any antiderivative of f, that is, F ’ =f.

60. Chapter 4, 4.4, P231 We noted that Part 1 can be rewritten as which says that if f is integrated and the result is then differentiated, we arrive back at the original function f.

61. Chapter 4, 4.4, P232 we define the average value of f on the interval [a,b] as

62. Chapter 4, 4.4, P233 THE MEAN VALUE THEOREM FOR INTEGRALS If f is continuous on [a,b], then there exists a number c in [a,b] such that that is,

63. Chapter 4, 4.4, P234 1.Let g(x)=, where f is the function whose graph is shown. (a) Evaluate g(0),g(1), g(2),g(3), and g(6). (b) On what interval is g increasing? (c) Where does g have a maximum value? (d) Sketch a rough graph of g.

64. Chapter 4, 4.4, P234 2.Let g(x)=, where f is the function whose graph is shown. (a) Evaluate g(x) for x=0,1,2,3,4,5, and 6. (b) Estimate g(7). (c) Where does g have a maximum value? Where does it have a minimum value? (d) Sketch a rough graph of g.

65. Chapter 4, 4.4, P235

67. Chapter 4, 4.5, P237 4. THE SUBSTITUTION RULE If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then

68. Chapter 4, 4.5, P239 5.THE SUBSTITUTION RULE FOR DEFINITE INTEGRALS If g ’ is continuous on [a,b] and f is continuous on the range of u=g(x), then

69. Chapter 4, 4.5, P240 6. INTEGRALS OF SYMMETRIC FUNCTIONS Suppose f is continuous on [-a,a]. (a)If f is even [f(-x)=f(x)], then (b)If f is odd [f(-x)=-f(x)], then

70. Chapter 4, 4.5, P240

72. Chapter 4, 4.5, P245 5. The following figure shows the graphs of f, f ’, and. Identify each graph, and explain your choices.