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# Lesson 24: The Definite Integral (Section 4 version)

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The limit of Riemann Sums has a name: the definite integral. We compute a few "easy" ones and show general properties.

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### Lesson 24: The Definite Integral (Section 4 version)

1. 1. Section 5.2 The Deﬁnite Integral V63.0121, Calculus I April 16, 2009 Announcements My ofﬁce is now WWH 624 Final Exam Friday, May 8, 2:00–3:50pm . . . . . .
2. 2. Outline Recall The deﬁnite integral as a limit Estimating the Deﬁnite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
3. 3. Cavalieri’s method in general Let f be a positive function deﬁned on the interval [a, b]. We want to ﬁnd the area between x = a, x = b, y = 0, and y = f(x). For each positive integer n, divide up the interval into n pieces. b−a . For each i between 1 and n, let xi be the ith Then ∆x = n step between a and b. So x0 = a b−a x1 = x0 + ∆x = a + n b−a x2 = x1 + ∆x = a + 2 · ... n b−a xi = a + i · ... n b−a xn = a + n · =b n .. . .. x . . 0 . 1 . . . . i . . .. n−1. n x x. x. x x . . . . . .
4. 4. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. left endpoints… n ∑ Ln = f(xi−1 )∆x i=1 ....... x . In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum n ∑ Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .
5. 5. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. right endpoints… n ∑ Rn = f(xi )∆x i=1 ....... x . In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum n ∑ Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .
6. 6. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. midpoints… ∑ ( xi−1 + xi ) n Mn = f ∆x 2 i=1 ....... x . In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum n ∑ Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .
7. 7. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. random points… ....... x . In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum n ∑ Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .
8. 8. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we made. . . . . . .
9. 9. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we made. . . . . . .
10. 10. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . . . x . matter what choice of ci we made. . . . . . .
11. 11. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . . . . x . matter what choice of ci we made. . . . . . .
12. 12. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . . . . . x . matter what choice of ci we made. . . . . . .
13. 13. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . . . . . . x . matter what choice of ci we made. . . . . . .
14. 14. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ....... x . matter what choice of ci we made. . . . . . .
15. 15. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ........ x . matter what choice of ci we made. . . . . . .
16. 16. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ......... x . matter what choice of ci we made. . . . . . .
17. 17. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .......... x . matter what choice of ci we made. . . . . . .
18. 18. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ........... x . matter what choice of ci we made. . . . . . .
19. 19. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............ x . matter what choice of ci we made. . . . . . .
20. 20. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............. x . matter what choice of ci we made. . . . . . .
21. 21. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .............. x . matter what choice of ci we made. . . . . . .
22. 22. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............... x . matter what choice of ci we made. . . . . . .
23. 23. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ................ x . matter what choice of ci we made. . . . . . .
24. 24. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ................. x . matter what choice of ci we made. . . . . . .
25. 25. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .................. x . matter what choice of ci we made. . . . . . .
26. 26. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ................... x . matter what choice of ci we made. . . . . . .
27. 27. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .................... x . matter what choice of ci we made. . . . . . .
28. 28. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ..................... . x . matter what choice of ci we made. . . . . . .
29. 29. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ...................... . x . matter what choice of ci we made. . . . . . .
30. 30. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ....................... . x . matter what choice of ci we made. . . . . . .
31. 31. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ........................ . x . matter what choice of ci we made. . . . . . .
32. 32. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ......................... . x . matter what choice of ci we made. . . . . . .
33. 33. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .......................... . x . matter what choice of ci we made. . . . . . .
34. 34. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ........................... . x . matter what choice of ci we made. . . . . . .
35. 35. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............................ . x . matter what choice of ci we made. . . . . . .
36. 36. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............................. . x . matter what choice of ci we made. . . . . . .
37. 37. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has ﬁnitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .............................. . x . matter what choice of ci we made. . . . . . .
38. 38. Outline Recall The deﬁnite integral as a limit Estimating the Deﬁnite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
39. 39. The deﬁnite integral as a limit Deﬁnition If f is a function deﬁned on [a, b], the deﬁnite integral of f from a to b is the number ∫b n ∑ f(x) dx = lim f(ci ) ∆x ∆x→0 a i=1 . . . . . .
40. 40. Notation/Terminology ∫ b f(x) dx a . . . . . .
41. 41. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) . . . . . .
42. 42. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) f(x) — integrand . . . . . .
43. 43. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) . . . . . .
44. 44. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) dx — ??? (a parenthesis? an inﬁnitesimal? a variable?) . . . . . .
45. 45. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) dx — ??? (a parenthesis? an inﬁnitesimal? a variable?) The process of computing an integral is called integration or quadrature . . . . . .
46. 46. The limit can be simpliﬁed Theorem If f is continuous on [a, b] or if f has only ﬁnitely many jump discontinuities, then f is integrable on [a, b]; that is, the deﬁnite ∫b integral f(x) dx exists. a . . . . . .
47. 47. The limit can be simpliﬁed Theorem If f is continuous on [a, b] or if f has only ﬁnitely many jump discontinuities, then f is integrable on [a, b]; that is, the deﬁnite ∫b integral f(x) dx exists. a Theorem If f is integrable on [a, b] then ∫ n ∑ b f(x) dx = lim f(xi )∆x, n→∞ a i=1 where b−a and xi = a + i ∆x ∆x = n . . . . . .
48. 48. Outline Recall The deﬁnite integral as a limit Estimating the Deﬁnite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
49. 49. Estimating the Deﬁnite Integral Given a partition of [a, b] into n pieces, let ¯i be the midpoint of x [xi−1 , xi ]. Deﬁne n ∑ Mn = f(¯i ) ∆x. x i=1 . . . . . .
50. 50. Example ∫ 1 4 dx using the midpoint rule and four divisions. Estimate 1 + x2 0 . . . . . .
51. 51. Example ∫ 1 4 dx using the midpoint rule and four divisions. Estimate 1 + x2 0 Solution 1 1 3 The partition is 0 < < < < 1, so the estimate is 4 2 4 ( ) 1 4 4 4 4 M4 = + + + 2 2 2 1 + (7/8)2 4 1 + (1/8) 1 + (3/8) 1 + (5/8) . . . . . .
52. 52. Example ∫ 1 4 dx using the midpoint rule and four divisions. Estimate 1 + x2 0 Solution 1 1 3 The partition is 0 < < < < 1, so the estimate is 4 2 4 ( ) 1 4 4 4 4 M4 = + + + 4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2 ( ) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64 . . . . . .
53. 53. Example ∫ 1 4 dx using the midpoint rule and four divisions. Estimate 1 + x2 0 Solution 1 1 3 The partition is 0 < < < < 1, so the estimate is 4 2 4 ( ) 1 4 4 4 4 M4 = + + + 4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2 ( ) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64 150, 166, 784 ≈ 3.1468 = 47, 720, 465 . . . . . .
54. 54. Outline Recall The deﬁnite integral as a limit Estimating the Deﬁnite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
55. 55. Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫b c dx = c(b − a) 1. a . . . . . .
56. 56. Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫b c dx = c(b − a) 1. a ∫ ∫ ∫ b b b [f(x) + g(x)] dx = f(x) dx + g(x) dx. 2. a a a . . . . . .
57. 57. Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫b c dx = c(b − a) 1. a ∫ ∫ ∫ b b b [f(x) + g(x)] dx = f(x) dx + g(x) dx. 2. a a a ∫ ∫ b b cf(x) dx = c f(x) dx. 3. a a . . . . . .
58. 58. Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫b c dx = c(b − a) 1. a ∫ ∫ ∫ b b b [f(x) + g(x)] dx = f(x) dx + g(x) dx. 2. a a a ∫ ∫ b b cf(x) dx = c f(x) dx. 3. a a ∫ ∫ ∫ b b b [f(x) − g(x)] dx = f(x) dx − g(x) dx. 4. a a a . . . . . .
59. 59. More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a . . . . . .
60. 60. More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a ∫ a f(x) dx = 0 a . . . . . .
61. 61. More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a ∫ a f(x) dx = 0 a This allows us to have ∫c ∫b ∫ c f(x) dx = f(x) dx + f(x) dx for all a, b, and c. 5. a a b . . . . . .
62. 62. Example Suppose f and g are functions with ∫4 f(x) dx = 4 0 ∫5 f(x) dx = 7 0 ∫5 g(x) dx = 3. 0 Find ∫5 [2f(x) − g(x)] dx (a) 0 ∫5 f(x) dx. (b) 4 . . . . . .
63. 63. Solution We have (a) ∫ ∫ ∫ 5 5 5 [2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx 0 0 0 = 2 · 7 − 3 = 11 . . . . . .
64. 64. Solution We have (a) ∫ ∫ ∫ 5 5 5 [2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx 0 0 0 = 2 · 7 − 3 = 11 (b) ∫ ∫ ∫ 5 5 4 f(x) dx − f(x) dx = f(x) dx 4 0 0 =7−4=3 . . . . . .
65. 65. Outline Recall The deﬁnite integral as a limit Estimating the Deﬁnite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
66. 66. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. . . . . . .
67. 67. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a . . . . . .
68. 68. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a 7. If f(x) ≥ g(x) for all x in [a, b], then ∫ ∫ b b f(x) dx ≥ g(x) dx a a . . . . . .
69. 69. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a 7. If f(x) ≥ g(x) for all x in [a, b], then ∫ ∫ b b f(x) dx ≥ g(x) dx a a 8. If m ≤ f(x) ≤ M for all x in [a, b], then ∫ b m(b − a) ≤ f(x) dx ≤ M(b − a) a . . . . . .
70. 70. Example ∫ 2 1 dx using the comparison properties. Estimate x 1 . . . . . .
71. 71. Example ∫ 2 1 dx using the comparison properties. Estimate x 1 Solution Since 1 1 ≤x≤ 2 1 for all x in [1, 2], we have ∫ 2 1 1 ·1≤ dx ≤ 1 · 1 x 2 1 . . . . . .