1. 9.1Concepts of Definite Integrals 9.2Finding Definite Integrals of Functions 9.3Further Techniques of Definite Integration Chapter Summary Case Study Definite Integrals 9 9.4Definite Integrals of Special Functions

2. P. 2 Suppose we want to find the area of the region bounded by the curve y  x 2, the x-axis and the vertical lines x  1 and x  3. Case Study As the bounded region is irregular, we cannot calculate the area by using a simple formula. Suppose we divide the interval 1  x  3 into four equal parts at the points 1.5, 2 and 2.5 as shown in the figure. My teacher said the solution is related to indefinite integrals... but I don’t know why. How can we calculate the area of the shaded region under the curve from x  1 to x  3? We then use the areas of the four rectangles to estimate the area under the curve. Each part has a width of unit  unit.

3. P. 3 In the figure, f (x) is a non-negative function in the interval a  x  b. 9.1 Concepts of Definite Integrals We can find the area of the region bounded by the curve y  f (x), the x-axis, the lines x  a and x  b by the following steps. 2.For i  1, 2,..., n, choose a point z i in the ith subinterval and draw a rectangle with height f (z i ) and width  x in the ith subinterval, as shown in the figure. A. Definition of Definite Integrals 1.Divide the interval a  x  b into n equal subintervals by the points x 0 (  a), x 1, x 2,..., x n (  b), such that each interval has a width of Hence the sum of the areas of the n rectangles is given by:

4. P. 4 Although the total area of the rectangles is just an approximation of the area of the bounded region, 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals Area under the curve  we can observe that when the width of each rectangle is getting smaller (as n approaches infinity), the sum of the areas of the rectangles becomes closer to the area under the curve. Here, we define the definite integral of f (x) as follows: Definition 9.1 If f (x) is defined in the interval a a  x  b, the definite integral of f (x) from a to b, which is denoted by, is defined as:

5. P. 5 Note: 1. From the above definition, we can observe the meaning behind the notation for definite integrals: ‘dx’ comes from ‘  x’ and the integral sign ‘   ’ is an elongated ‘S’ which means ‘summation’. 2.In the definite integral, a and b are called the lower limit and the upper limit respectively. 3.Note that the infinite sum is equal to the area under the curve only if the curve is non-negative in the interval. In next chapter, we will learn how to find the area under the curve by using definite integral. 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals In addition, the limit sum is closely related to the indefinite integral. This will be explained in the next section.

6. P. 6 Example 9.1T Solution: First we divide the interval 0  x  1 into n subintervals of width  x. 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals Using the identity evaluate Thus, If we choose z i as the right end point of each subinterval, then we have

7. P. 7 Example 9.2T Using the identity sin  + sin 2  + … + evaluate First we divide the interval 0  x  1 into n subintervals of width  x. 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals Thus, Solution:

8. P. 8 Example 9.2T Solution: 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals Using the identity sin  + sin 2  + … + evaluate

9. P. 9 From the above examples, we observe that the definite integral is a real number which is independent of the variable x. We say that x is a dummy variable and it can be replaced by another letter, say u, without changing the value of the integral. 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals In other words,. As the two integrals describe the same graph, the only difference is whether the horizontal axis is labelled ‘x-axis’ or ‘u-axis’. So their corresponding areas are the same and the definite integrals also have the same value.

10. P. 10 In this section, we will study the properties of definite integrals. Before that, let us introduce the following definitions of definite integral first: 9.1 Concepts of Definite Integrals Now let us consider three useful properties for evaluating integrals: B. Properties of Definite Integrals Definition 9.2 Let f (x) be a continuous function in the interval a  x  b. Then we define (a) (b) Properties of Definite Integrals Let f (x) and g(x) be continuous functions in the interval a  x  b. 9.1. 9.2. 9.3.

11. P. 11 Example 9.3T Solution: Evaluate 9.1 Concepts of Definite Integrals B. Properties of Definite Integrals (a) (b)

12. P. 12 In the last section, we saw that it is very tedious and time consuming to evaluate a definite integral from the definition. In this section, we will introduce the Fundamental Theorem of Calculus, which enables us to evaluate a definite integral more efficiently. 9.2 Finding Definite Integrals of Functions Functions Theorem 9.1 Fundamentals Theorem of Calculus If f (x) be a continuous function in the interval a  x  b and F (x) is a primitive function of f (x), then

13. P. 13 Proof: For a  x  b, let be the area of the region enclosed by the curve y  f (t), the t-axis, the vertical lines t  a and t  x as shown in the figure. Then 9.2 Finding Definite Integrals of Functions Functions Now, when h  0, area of PQRT  area of PQRS, that is, Thus,  f (x) ∴ A(x) is a primitive function of f (x).

14. P. 14 Since F(x) is also a primitive function of f (x), it differs from A(x) by just a constant, for example C. Then we have A(x)  F(x) + C............(*) 9.2 Finding Definite Integrals of Functions Functions Substitute x  a into (*), ∴ A(x)  F(x) – F(a)............(**) ∴ C  –F(a) Substitute x  b into (**), A(b)  F(b) – F(a) When applying the above theorem, for simplicity, we use the notation to denote F(b) – F(a).

15. P. 15 Example 9.4T Solution: Since 9.2 Finding Definite Integrals of Functions Functions Evaluate is a primitive function of x 2 + 4x. By the Fundamental Theorem of Calculus,

16. P. 16 Example 9.5T 9.2 Finding Definite Integrals of Functions Functions Solution: Evaluate

17. P. 17 Example 9.6T 9.2 Finding Definite Integrals of Functions Functions Solution: Evaluate

18. P. 18 Example 9.7T 9.2 Finding Definite Integrals of Functions Functions Solution: Evaluate

19. P. 19 Example 9.8T 9.2 Finding Definite Integrals of Functions Functions Solution: Evaluate

20. P. 20 Example 9.9T 9.2 Finding Definite Integrals of Functions Functions Solution: (a)(a) (b)Hence evaluate (a)Find (b)(b) By (a)

21. P. 21 Similar technique can also be applied when evaluating definite integrals. In the last chapter, we have learnt the method of integration by substitution for indefinite integrals. 9.3 Further Techniques of Definite Integration Integration Theorem 9.2 Integration by Substitution Let u  g(x) be a differentiable function in the interval a  x  b. If y  f(u) is a continuous function in the interval g(a)  u  g(b), then we have A. Integration by Substitution

22. P. 22 Proof: By the Fundamental Theorem of Calculus,  f(u)g'(x) f(u)g'(x) 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution

23. P. 23 Example 9.10T 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution Solution: Let u  ln x. Then Evaluate When x  e, u  1. When x  e 2, u  2.

24. P. 24 Example 9.11T 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution Solution: Evaluate Let u  tan x. Then du  sec 2 xdx. When x  0, u  0. When u  1.

25. P. 25 In the above example, we can see that after substitution, the upper limit may become smaller than the lower limit. This is the reason why we need to learn Definition 9.2(b). 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution

26. P. 26 9.3 Further Techniques of Definite Integration Integration Let x be a real number. 1.sin –1 x is defined as the angle  such that sin   x (where –1  x  1) and 2. cos –1 x is defined as the angle  such that cos   x (where –1  x  1) and 0    . 3. tan –1 x is defined as the angle  such that tan   x and A. Integration by Substitution When using trigonometric substitution x  f (  ), the upper and lower limits of x have to be changed to that of , and the range of  follows that of f –1 (inverse function of f ) as shown below: We can also use trigonometric substitution to evaluate definite integrals as in Chapter 8.

27. P. 27 Example 9.12T 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution Solution: Evaluate Let x  tan . Then When x  0,   0. When x  1,

28. P. 28 Example 9.13T 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution Solution: Given that evaluate sin(  – x)  sin x, cos(  – x)  –cos x

29. P. 29 Example 9.13T 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution Solution: Given that evaluate For, let When x  0,.When x  ,.

30. P. 30 9.3 Further Techniques of Definite Integration Integration B. Integration by Parts Theorem 9.3 Integration by Parts Let u and v be two differentiable functions. Then, In Chapter 8, we have already learnt the method of integration by parts for indefinite integrals. For definite integrals, we can also apply a similar method to evaluate integrals and it is stated as follows: We have already proved the corresponding theorem in the last chapter.

31. P. 31 Example 9.14T 9.3 Further Techniques of Definite Integration Integration B. Integration by Parts Solution: Evaluate

32. P. 32 Example 9.15T 9.3 Further Techniques of Definite Integration Integration B. Integration by Parts Solution: Evaluate

33. P. 33 Example 9.16T 9.3 Further Techniques of Definite Integration Integration B. Integration by Parts Solution: Evaluate

34. P. 34 In Chapter 7, we learnt that for a function f (x), (i)if f (–x)  f (x), then it is called an even function, and (ii)if f (–x)  –f (x), then it is called an odd function. 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions For these functions, we have the following theorems for their definite integrals: Theorem 9.4 Definite Integrals of Odd and Even Functions Let k be a constant and f (x) be a continuous function in the interval –k  x  k. (a)If f (x) is an even function, then. (b)If f (x) is an odd function, then.

35. P. 35 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions Proof: Case 1: f (x) is an even function Equation (*) becomes: When x  –k, u  k. When x  0, u  0. Let u  – x. Then du  –dx. Case 2: f (x) is an odd function Equation (*) becomes:  0

36. P. 36 The above theorem can be explained by the area under a curve. 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions For an even function, the curve is symmetrical about the y-axis as shown in the figure. Therefore, the areas under the curve on the L.H.S. and R.H.S. of the y-axis should be the same. The total area is twice that on the R.H.S. of the y-axis, as stated in the theorem. In the case of an odd function, the curve is symmetrical about the origin as shown in the figure. So the integrals on both sides cancel each other and give a sum of zero. Therefore, the areas under the curve on the L.H.S. and R.H.S. of the y-axis are the same, but they are in opposite signs.

37. P. 37 Example 9.17T 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions Solution: Evaluate  f (x) is an odd function.

38. P. 38 Example 9.18T 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions Solution: Evaluate Let f (x) = tan x sec x and g(x) = sec 2 x.  f (x) is an odd function.  g(x) is an even function.

39. P. 39 Example 9.19T 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions Solution: Evaluate  f (x) is an even function.

40. P. 40 As we have learnt, the graph of y = sin x repeats itself at intervals of 2  as shown in the figure. 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions This means that sin (x + 2  ) = sin x for all x. Thus we say that y = sin x is a periodic function with a period of 2 . Definition 9.3 A function f (x) is said to be a periodic function if there is a positive value T, such as f (x+T) = f (x) for real values of x. in this case, the smallest value of T which satisfies the relationship above is called the period of f (x).

41. P. 41 For the definite integrals of periodic functions, we have the following theorem: 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions Theorem 9.5 Definite Integrals of Periodic Functions If f (x) is a periodic function with period T, then for all real constants k. Theorem 9.5 tells us that if a function f (x) is periodic with period T, then the definite integral of f (x) over an interval of length T is a constant, regardless of the position of the interval.

42. P. 42 The following is the proof of the theorem: 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions Proof: Let u  x – T. Then du  dx. When x  T, u  0. When x  k + T, u  k.

43. P. 43 Example 9.20T Solution: 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions Let f (x) be a periodic function with period 5 and evaluate.

44. P. 44 Example 9.21T Solution: 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions Let f (x) be a function and, where k is a constant. If and g(x) is a periodic function with period T, show that Let x = x 0 where x 0 is an arbitrary real constant.

45. P. 45 Solution: 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions  g(x) is a periodic function with period T. Example 9.21T Let f (x) be a function and, where k is a constant. If and g(x) is a periodic function with period T, show that

46. P. 46 Example 9.22T Solution: 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions Let f (x) be an even and periodic function with period 5. If find the value of k such that Let k = 5m + n, where m is a positive integer and 0  n  5.

47. P. 47 Example 9.22T Solution: 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions For m = 8, n = 0, L.H.S.  m = 8, n = 0 are solutions of the equation. Let f (x) be an even and periodic function with period 5. If find the value of k such that

48. P. 48 9.1 Concepts of Definite Integrals Chapter Summary 1.If f (x) be a function defined in the interval a a  x  b. The definite integral of f (x) from a to b is given by where and z i is a point in the subinterval a + (i – 1)  x  x  a + i  x. 2.Let f (x) and g(x) be continuous functions in the interval a a  x  b. Then

49. P. 49 9.2 Finding Definite Integrals of Functions If f (x) is a continuous function in the interval a a  x  b and, then Chapter Summary

50. P. 50 1.Let u  g(x) be a differentiable function in the interval a  x  b. If y  f(u) is a continuous function in the interval g(a)  u  g(b), then 2.Let u and v be differentiable functions. Then, 9.3 Further Techniques of Definite Integration Chapter Summary

51. P. 51 1.Let f (x) be a continuous function and k be a constant. 2.If f (x) is a periodic function with period T, then 9.4 Definite Integrals of Special Functions Chapter Summary (a) If f (x) is an even function, then (b) If f (x) is an odd function, then (i) for all real constants k, (ii) for all positive integers n.

52. Follow-up 9.1 Solution: First we divide the interval 0  x  1 into n subintervals of width  x. Using the identity evaluate 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals Thus, If we choose z i as the right end point of each subinterval, then we have

53. Follow-up 9.2 Solution: First we divide the interval into n subintervals of width  x. Using the identity cos  + cos 2  + … + evaluate 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals Thus,

54. Follow-up 9.2 9.1 Concepts of Definite Integrals A. Definition of Definite Integrals Using the identity cos  + cos 2  + … + evaluate Solution:

55. Follow-up 9.3 Solution: evaluate (a) 9.1 Concepts of Definite Integrals B. Properties of Definite Integrals (b)

56. Follow-up 9.4 Solution: Since Evaluate 9.2 Finding Definite Integrals of Functions Functions is a primitive function of x 3 + 4x + 1. By the Fundamental Theorem of Calculus,

57. Solution: Follow-up 9.5 Evaluate 9.2 Finding Definite Integrals of Functions Functions

58. Follow-up 9.6 Solution: Find 9.2 Finding Definite Integrals of Functions Functions

59. Follow-up 9.7 Solution: Evaluate 9.2 Finding Definite Integrals of Functions Functions

60. Follow-up 9.8 Solution: Evaluate 9.2 Finding Definite Integrals of Functions Functions

61. (a)(a) (b)Hence evaluate (a)Find Solution: 9.2 Finding Definite Integrals of Functions Functions Follow-up 9.9

62. (b)By (a), (b)Hence evaluate (a)Find Solution: 9.2 Finding Definite Integrals of Functions Functions

63. Follow-up 9.10 Solution: Let u  x 3 + 1. Then Evaluate 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution When x  0, u  0 3 + 1  1. When x  2, u  2 3 + 1  9.

64. Follow-up 9.11 Solution: Let u  csc x. Then du  –csc x cot xdx. Evaluate 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution

65. Follow-up 9.12 Solution: Let x  sec . Then dx  sec  tan  d  Evaluate 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution

66. Follow-up 9.13 (a)Let x  2 – u. Then dx  –du. (b)Hence evaluate (a)Let f (x) be a continuous function. Show that Solution: 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution When x  0, u  2. When x  2, u  0. u and x are dummy variables

67. Follow-up 9.13 (b)(b) (b)Hence evaluate (a)Let f (x) be a continuous function. Show that Solution: 9.3 Further Techniques of Definite Integration Integration A. Integration by Substitution

68. Follow-up 9.14 Solution: Evaluate 9.3 Further Techniques of Definite Integration Integration B. Integration by Parts

69. Follow-up 9.15 Solution: Evaluate 9.3 Further Techniques of Definite Integration Integration B. Integration by Parts

70. Follow-up 9.16 Solution: Evaluate 9.3 Further Techniques of Definite Integration Integration B. Integration by Parts

71. Follow-up 9.17 Solution: Evaluate 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions  f (x) is an odd function.

72. Follow-up 9.18 Solution: Evaluate 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions Let f (x) = x 3 + 2x and g(x) = 7 – 6x 2.  f (x) is an odd function.  g(x) is an even function.

73. Follow-up 9.19 Solution: (a)(a) Evaluate the following definite integrals. (a)(b) 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions  f (x) is an even function.

74. Follow-up 9.19 (a)(a) Evaluate the following definite integrals. (a)(b) 9.4 Definite Integrals of Special Functions A. Definite Integrals of Even Functions and Odd Functions Odd Functions (b)(b)  g(x) is an odd function. Solution:

75. Follow-up 9.20 Solution: Let f (x) be an even and periodic function with period 2. If evaluate 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions

76. Follow-up 9.21 Solution: Let f (x) be a function. If g(x) is a periodic function with period and, where k is a constant, show that 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions Let x = x 0, where x 0 is an arbitrary real constant.

77. Follow-up 9.21 Solution: Let f (x) be a function. If g(x) is a periodic function with period and, where k is a constant, show that 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions ∵ g(x) is a periodic function with period.

78. Follow-up 9.22 9.4 Definite Integrals of Special Functions B. Definite Integrals of Periodic Functions Solution: Let f (x) be an even and periodic function with period 5. If evaluate