Lesson 24: Areas and Distances, The Definite Integral (slides)

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We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.

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Lesson 24: Areas and Distances, The Definite Integral (slides)

  1. 1. Sec on 5.1–5.2 Areas and Distances, The Definite Integral V63.0121.011: Calculus I Professor Ma hew Leingang New York University April 25, 2011.
  2. 2. Announcements Quiz 5 on Sec ons 4.1–4.4 April 28/29 Final Exam Thursday May 12, 2:00–3:50pm cumula ve loca on TBD old exams on common website
  3. 3. Objectives from Section 5.1 Compute the area of a region by approxima ng it with rectangles and le ng the size of the rectangles tend to zero. Compute the total distance traveled by a par cle by approxima ng it as distance = (rate)( me) and le ng the me intervals over which one approximates tend to zero.
  4. 4. Objectives from Section 5.2 Compute the definite integral using a limit of Riemann sums Es mate the definite integral using a Riemann sum (e.g., Midpoint Rule) Reason with the definite integral using its elementary proper es.
  5. 5. Outline Area through the Centuries Euclid Archimedes Cavalieri Generalizing Cavalieri’s method Analogies Distances Other applica ons The definite integral as a limit Es ma ng the Definite Integral Proper es of the integral Comparison Proper es of the Integral
  6. 6. Easy Areas: Rectangle Defini on The area of a rectangle with dimensions ℓ and w is the product A = ℓw. w . ℓ It may seem strange that this is a defini on and not a theorem but we have to start somewhere.
  7. 7. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. . b
  8. 8. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. h . b
  9. 9. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. h .
  10. 10. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. h . b
  11. 11. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. h . So b Fact The area of a parallelogram of base width b and height h is A = bh
  12. 12. Easy Areas: Triangle By copying and pas ng, a triangle can be made into a parallelogram. . b
  13. 13. Easy Areas: Triangle By copying and pas ng, a triangle can be made into a parallelogram. h . b
  14. 14. Easy Areas: Triangle By copying and pas ng, a triangle can be made into a parallelogram. h . So b Fact The area of a triangle of base width b and height h is 1 A = bh 2
  15. 15. Easy Areas: Other Polygons Any polygon can be triangulated, so its area can be found by summing the areas of the triangles: . .
  16. 16. Hard Areas: Curved Regions . ???
  17. 17. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (a er Euclid) Geometer Weapons engineer
  18. 18. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (a er Euclid) Geometer Weapons engineer
  19. 19. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (a er Euclid) Geometer Weapons engineer
  20. 20. Archimedes and the Parabola . Archimedes found areas of a sequence of triangles inscribed in a parabola. A=
  21. 21. Archimedes and the Parabola 1 . Archimedes found areas of a sequence of triangles inscribed in a parabola. A=1
  22. 22. Archimedes and the Parabola 1 1 1 8 8 . Archimedes found areas of a sequence of triangles inscribed in a parabola. 1 A=1+2· 8
  23. 23. Archimedes and the Parabola 1 1 64 64 1 1 1 8 8 1 1 64 64 . Archimedes found areas of a sequence of triangles inscribed in a parabola. 1 1 A=1+2· +4· + ··· 8 64
  24. 24. Archimedes and the Parabola 1 1 64 64 1 1 1 8 8 1 1 64 64 . Archimedes found areas of a sequence of triangles inscribed in a parabola. 1 1 1 1 1 A=1+2· +4· + ··· = 1 + + + ··· + n + ··· 8 64 4 16 4
  25. 25. Summing the series We need to know the value of the series 1 1 1 1+ + + ··· + n + ··· 4 16 4
  26. 26. Summing a geometric series Fact For any number r and any posi ve integer n, (1 − r)(1 + r + r2 + · · · + rn ) = 1 − rn+1 .
  27. 27. Summing a geometric series Fact For any number r and any posi ve integer n, (1 − r)(1 + r + r2 + · · · + rn ) = 1 − rn+1 . Proof. (1 − r)(1 + r + r2 + · · · + rn ) = (1 + r + r2 + · · · + rn ) − r(1 + r + r2 + · · · + rn ) = (1 + r + r2 + · · · + rn ) − (r + r2 + r3 · · · + rn + rn+1 ) = 1 − rn+1
  28. 28. Summing a geometric series Fact For any number r and any posi ve integer n, (1 − r)(1 + r + r2 + · · · + rn ) = 1 − rn+1 . Corollary 1 − rn+1 1 + r + ··· + r =n 1−r
  29. 29. Summing the series We need to know the value of the series 1 1 1 1+ + + ··· + n + ··· 4 16 4 Using the corollary, 1 1 1 1 − (1/4)n+1 1+ + + ··· + n = 4 16 4 1 − 1/4
  30. 30. Summing the series We need to know the value of the series 1 1 1 1+ + + ··· + n + ··· 4 16 4 Using the corollary, 1 1 1 1 − (1/4)n+1 1 4 1+ + + ··· + n = → 3 = as n → ∞. 4 16 4 1 − 1/4 /4 3
  31. 31. Cavalieri Italian, 1598–1647 Revisited the area problem with a different perspec ve
  32. 32. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: . 0 1
  33. 33. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: 1 L2 = 8 . 0 1 1 2
  34. 34. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: 1 L2 = 8 L3 = . 0 1 2 1 3 3
  35. 35. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 . 0 1 2 1 3 3
  36. 36. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 L4 = . 0 1 2 3 1 4 4 4
  37. 37. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 1 4 9 14 L4 = + + = . 64 64 64 64 0 1 2 3 1 4 4 4
  38. 38. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 1 4 9 14 L4 = + + = . 64 64 64 64 0 1 2 3 4 1 L5 = 5 5 5 5
  39. 39. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 1 4 9 14 L4 = + + = . 64 64 64 64 1 4 9 16 30 0 1 2 3 4 1 L5 = + + + = 125 125 125 125 125 5 5 5 5
  40. 40. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 1 4 9 14 L4 = + + = . 64 64 64 64 1 4 9 16 30 0 1 L5 = + + + = 125 125 125 125 125 Ln =?
  41. 41. What is Ln? 1 Divide the interval [0, 1] into n pieces. Then each has width . n
  42. 42. What is Ln? 1 Divide the interval [0, 1] into n pieces. Then each has width . The n rectangle over the ith interval and under the parabola has area ( )2 1 i−1 (i − 1)2 · = . n n n3
  43. 43. What is Ln? 1 Divide the interval [0, 1] into n pieces. Then each has width . The n rectangle over the ith interval and under the parabola has area ( )2 1 i−1 (i − 1)2 · = . n n n3 So 1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2 Ln = 3 + 3 + · · · + = n n n3 n3
  44. 44. The Square Pyramidial Numbers Fact Let n be a posi ve integer. Then n(n − 1)(2n − 1) 1 + 22 + 32 + · · · + (n − 1)2 = 6 This formula was known to the Arabs and discussed by Fibonacci in his book Liber Abaci.
  45. 45. What is Ln? 1 Divide the interval [0, 1] into n pieces. Then each has width . The n rectangle over the ith interval and under the parabola has area ( )2 1 i−1 (i − 1)2 · = . n n n3 So 1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2 Ln = 3 + 3 + · · · + = n n n3 n3 So n(n − 1)(2n − 1) Ln = 6n3
  46. 46. What is Ln? 1 Divide the interval [0, 1] into n pieces. Then each has width . The n rectangle over the ith interval and under the parabola has area ( )2 1 i−1 (i − 1)2 · = . n n n3 So 1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2 Ln = 3 + 3 + · · · + = n n n3 n3 So n(n − 1)(2n − 1) 1 Ln = → 6n3 3 as n → ∞.
  47. 47. Cavalieri’s method for different functions Try the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n
  48. 48. Cavalieri’s method for different functions Try the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n 1 1 1 23 1 (n − 1)3 = · 3 + · 3 + ··· + · n n n n n n3
  49. 49. Cavalieri’s method for different functions Try the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n 1 1 1 23 1 (n − 1)3 = · 3 + · 3 + ··· + · n n n n n n3 1 + 23 + 33 + · · · + (n − 1)3 = n4
  50. 50. Nicomachus’s Theorem Fact (Nicomachus 1st c. CE, Aryabhata 5th c., Al-Karaji 11th c.) 1 + 23 + 33 + · · · + (n − 1)3 = [1 + 2 + · · · + (n − 1)]2 [1 ]2 = 2 n(n − 1)
  51. 51. Cavalieri’s method for different functions Try the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n 1 1 1 23 1 (n − 1)3 = · 3 + · 3 + ··· + · n n n n n n3 1 + 23 + 33 + · · · + (n − 1)3 = n4 n2 (n − 1)2 = 4n4
  52. 52. Cavalieri’s method for different functions Try the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n 1 1 1 23 1 (n − 1)3 = · 3 + · 3 + ··· + · n n n n n n3 1 + 23 + 33 + · · · + (n − 1)3 = n4 n2 (n − 1)2 1 = → 4n4 4 as n → ∞.
  53. 53. Cavalieri’s method with different heights 1 13 1 23 1 n3 Rn = · 3 + · 3 + ··· + · 3 n n n n n n 1 + 2 + 3 + ··· + n 3 3 3 3 = n4 1 [ ]2 = 4 1 n(n + 1) n 2 n2 (n + 1)2 1 . = → 4n4 4 as n → ∞.
  54. 54. Cavalieri’s method with different heights 1 13 1 23 1 n3 Rn = · 3 + · 3 + ··· + · 3 n n n n n n 1 + 2 + 3 + ··· + n 3 3 3 3 = n4 1 [ ]2 = 4 1 n(n + 1) n 2 n2 (n + 1)2 1 . = → 4n4 4 as n → ∞. So even though the rectangles overlap, we s ll get the same answer.
  55. 55. Outline Area through the Centuries Euclid Archimedes Cavalieri Generalizing Cavalieri’s method Analogies Distances Other applica ons The definite integral as a limit Es ma ng the Definite Integral Proper es of the integral Comparison Proper es of the Integral
  56. 56. Cavalieri’s method in general Problem Let f be a posi ve func on defined on the interval [a, b]. Find the area between x = a, x = b, y = 0, and y = f(x). . . x x x0 x1. . . xi . . xn−1 n
  57. 57. Cavalieri’s method in general For each posi ve integer n, divide up the interval into n pieces. Then b−a ∆x = . For each i between 1 and n, let xi be the ith step n between a and b. 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 . x x x0 x1. . . xi . . xn−1 n xn = a + n · =b n
  58. 58. Forming Riemann Sums 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 Thus we approximate area under a curve by a sum of areas of rectangles.
  59. 59. Forming Riemann sums We have many choices of representa ve points to approximate the area in each subinterval. le endpoints… ∑ n Ln = f(xi−1 )∆x i=1 . x
  60. 60. Forming Riemann sums We have many choices of representa ve points to approximate the area in each subinterval. right endpoints… ∑ n Rn = f(xi )∆x i=1 . x
  61. 61. Forming Riemann sums We have many choices of representa ve points to approximate the area in each subinterval. midpoints… ∑ ( xi−1 + xi ) n Mn = f ∆x i=1 2 . x
  62. 62. Forming Riemann sums We have many choices of representa ve points to approximate the area in each subinterval. the maximum value on the interval… ∑ n Un = max {f(x)} ∆x xi−1 ≤x≤xi i=1 . x
  63. 63. Forming Riemann sums We have many choices of representa ve points to approximate the area in each subinterval. the minimum value on the interval… ∑ n Ln = min {f(x)} ∆x xi−1 ≤x≤xi i=1 . x
  64. 64. Forming Riemann sums We have many choices of representa ve points to approximate the area in each subinterval. …even random points! . x
  65. 65. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make.
  66. 66. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make.
  67. 67. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L1 = 3.0 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  68. 68. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L2 = 5.25 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  69. 69. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L3 = 6.0 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  70. 70. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L4 = 6.375 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  71. 71. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L5 = 6.59988 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  72. 72. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L6 = 6.75 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  73. 73. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L7 = 6.85692 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  74. 74. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L8 = 6.9375 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  75. 75. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L9 = 6.99985 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  76. 76. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L10 = 7.04958 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  77. 77. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L11 = 7.09064 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  78. 78. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L12 = 7.125 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  79. 79. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L13 = 7.15332 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  80. 80. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L14 = 7.17819 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  81. 81. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L15 = 7.19977 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  82. 82. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L16 = 7.21875 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  83. 83. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L17 = 7.23508 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  84. 84. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L18 = 7.24927 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  85. 85. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L19 = 7.26228 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  86. 86. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L20 = 7.27443 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  87. 87. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L21 = 7.28532 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  88. 88. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L22 = 7.29448 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  89. 89. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L23 = 7.30406 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  90. 90. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L24 = 7.3125 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  91. 91. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L25 = 7.31944 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  92. 92. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L26 = 7.32559 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  93. 93. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L27 = 7.33199 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  94. 94. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L28 = 7.33798 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  95. 95. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L29 = 7.34372 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  96. 96. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L30 = 7.34882 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. le endpoints
  97. 97. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R1 = 12.0 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  98. 98. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R2 = 9.75 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  99. 99. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R3 = 9.0 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  100. 100. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R4 = 8.625 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  101. 101. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R5 = 8.39969 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  102. 102. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R6 = 8.25 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  103. 103. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R7 = 8.14236 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  104. 104. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R8 = 8.0625 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  105. 105. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R9 = 7.99974 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  106. 106. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R10 = 7.94933 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  107. 107. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R11 = 7.90868 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  108. 108. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R12 = 7.875 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  109. 109. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R13 = 7.84541 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  110. 110. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R14 = 7.8209 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  111. 111. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R15 = 7.7997 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  112. 112. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R16 = 7.78125 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  113. 113. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R17 = 7.76443 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  114. 114. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R18 = 7.74907 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  115. 115. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R19 = 7.73572 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  116. 116. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R20 = 7.7243 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  117. 117. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R21 = 7.7138 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  118. 118. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R22 = 7.70335 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  119. 119. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R23 = 7.69531 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  120. 120. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R24 = 7.6875 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  121. 121. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R25 = 7.67934 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  122. 122. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R26 = 7.6715 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  123. 123. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R27 = 7.66508 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  124. 124. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R28 = 7.6592 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  125. 125. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R29 = 7.65388 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  126. 126. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump R30 = 7.64864 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. right endpoints
  127. 127. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M1 = 7.5 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  128. 128. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M2 = 7.5 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  129. 129. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M3 = 7.5 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  130. 130. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M4 = 7.5 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  131. 131. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M5 = 7.4998 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  132. 132. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M6 = 7.5 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  133. 133. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M7 = 7.4996 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  134. 134. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M8 = 7.5 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  135. 135. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M9 = 7.49977 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  136. 136. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M10 = 7.49947 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  137. 137. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M11 = 7.49966 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  138. 138. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M12 = 7.5 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  139. 139. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M13 = 7.49937 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  140. 140. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M14 = 7.49954 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  141. 141. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M15 = 7.49968 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  142. 142. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M16 = 7.49988 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  143. 143. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M17 = 7.49974 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  144. 144. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M18 = 7.49916 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  145. 145. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M19 = 7.49898 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  146. 146. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M20 = 7.4994 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  147. 147. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M21 = 7.49951 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  148. 148. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M22 = 7.49889 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  149. 149. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M23 = 7.49962 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  150. 150. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M24 = 7.5 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  151. 151. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M25 = 7.49939 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  152. 152. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M26 = 7.49847 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  153. 153. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M27 = 7.4985 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  154. 154. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M28 = 7.4986 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  155. 155. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M29 = 7.49878 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  156. 156. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump M30 = 7.49872 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. midpoints
  157. 157. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U1 = 12.0 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  158. 158. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U2 = 10.55685 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  159. 159. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U3 = 10.0379 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  160. 160. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U4 = 9.41515 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  161. 161. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U5 = 8.96004 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  162. 162. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U6 = 8.76895 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  163. 163. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U7 = 8.6033 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  164. 164. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U8 = 8.45757 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  165. 165. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U9 = 8.34564 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  166. 166. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U10 = 8.27084 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  167. 167. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U11 = 8.20132 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  168. 168. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U12 = 8.13838 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  169. 169. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U13 = 8.0916 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  170. 170. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U14 = 8.05139 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  171. 171. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U15 = 8.01364 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  172. 172. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U16 = 7.98056 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  173. 173. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U17 = 7.9539 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  174. 174. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U18 = 7.92815 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  175. 175. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U19 = 7.90414 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  176. 176. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U20 = 7.88504 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  177. 177. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U21 = 7.86737 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  178. 178. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U22 = 7.84958 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  179. 179. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U23 = 7.83463 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  180. 180. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U24 = 7.82187 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  181. 181. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U25 = 7.80824 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  182. 182. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U26 = 7.79504 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  183. 183. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U27 = 7.78429 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  184. 184. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U28 = 7.77443 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  185. 185. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U29 = 7.76495 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  186. 186. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump U30 = 7.7558 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. maximum points
  187. 187. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L1 = 3.0 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  188. 188. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L2 = 4.44312 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  189. 189. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L3 = 4.96208 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  190. 190. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L4 = 5.58484 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  191. 191. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L5 = 6.0395 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  192. 192. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L6 = 6.23103 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  193. 193. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L7 = 6.39577 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  194. 194. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L8 = 6.54242 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  195. 195. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L9 = 6.65381 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  196. 196. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L10 = 6.72797 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  197. 197. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L11 = 6.7979 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  198. 198. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L12 = 6.8616 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  199. 199. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L13 = 6.90704 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  200. 200. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L14 = 6.94762 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  201. 201. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L15 = 6.98575 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  202. 202. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L16 = 7.01942 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  203. 203. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L17 = 7.04536 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  204. 204. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L18 = 7.07005 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  205. 205. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L19 = 7.09364 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  206. 206. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L20 = 7.1136 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  207. 207. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L21 = 7.13155 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  208. 208. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L22 = 7.14804 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  209. 209. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L23 = 7.16441 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  210. 210. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L24 = 7.17812 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  211. 211. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L25 = 7.19025 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  212. 212. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L26 = 7.2019 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  213. 213. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L27 = 7.21265 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  214. 214. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L28 = 7.22269 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  215. 215. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L29 = 7.23251 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  216. 216. Theorem of the Day Theorem If f is a con nuous func on on [a, b] or has finitely many jump L30 = 7.24162 discon nui es, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x ma er what choice of ci we make. minimum points
  217. 217. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4)
  218. 218. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4) Want the slope of a curve
  219. 219. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4) Want the area of a curved Want the slope of a curve region
  220. 220. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4) Want the area of a curved Want the slope of a curve region Only know the slope of lines
  221. 221. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4) Want the area of a curved Want the slope of a curve region Only know the slope of Only know the area of lines polygons
  222. 222. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4) Want the area of a curved Want the slope of a curve region Only know the slope of Only know the area of lines polygons Approximate curve with a line
  223. 223. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4) Want the area of a curved Want the slope of a curve region Only know the slope of Only know the area of lines polygons Approximate curve with a Approximate region with line polygons
  224. 224. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4) Want the area of a curved Want the slope of a curve region Only know the slope of Only know the area of lines polygons Approximate curve with a Approximate region with line polygons Take limit over be er and Take limit over be er and be er approxima ons be er approxima ons
  225. 225. Outline Area through the Centuries Euclid Archimedes Cavalieri Generalizing Cavalieri’s method Analogies Distances Other applica ons The definite integral as a limit Es ma ng the Definite Integral Proper es of the integral Comparison Proper es of the Integral
  226. 226. Distances Just like area = length × width, we have distance = rate × me. So here is another use for Riemann sums.
  227. 227. Application: Dead Reckoning
  228. 228. Computing position by Dead Reckoning Example A sailing ship is cruising back and forth along a channel (in a straight line). At noon the ship’s posi on and velocity are recorded, but shortly therea er a storm blows in and posi on is impossible to measure. The velocity con nues to be recorded at thirty-minute intervals.
  229. 229. Computing position by Dead Reckoning Example Time 12:00 12:30 1:00 1:30 2:00 Speed (knots) 4 8 12 6 4 Direc on E E E E W Time 2:30 3:00 3:30 4:00 Speed 3 3 5 9 Direc on W E E E Es mate the ship’s posi on at 4:00pm.
  230. 230. Solution Solu on We es mate that the speed of 4 knots (nau cal miles per hour) is maintained from 12:00 un l 12:30. So over this me interval the ship travels ( )( ) 4 nmi 1 hr = 2 nmi hr 2 We can con nue for each addi onal half hour and get distance = 4 × 1/2 + 8 × 1/2 + 12 × 1/2 + 6 × 1/2 − 4 × 1/2 − 3 × 1/2 + 3 × 1/2 + 5 × 1/2 = 15.5 So the ship is 15.5 nmi east of its original posi on.
  231. 231. Analysis This method of measuring posi on by recording velocity was necessary un l global-posi oning satellite technology became widespread If we had velocity es mates at finer intervals, we’d get be er es mates. If we had velocity at every instant, a limit would tell us our exact posi on rela ve to the last me we measured it.
  232. 232. Other uses of Riemann sums Anything with a product! Area, volume Anything with a density: Popula on, mass Anything with a “speed:” distance, throughput, power Consumer surplus Expected value of a random variable
  233. 233. Outline Area through the Centuries Euclid Archimedes Cavalieri Generalizing Cavalieri’s method Analogies Distances Other applica ons The definite integral as a limit Es ma ng the Definite Integral Proper es of the integral Comparison Proper es of the Integral
  234. 234. The definite integral as a limit Defini on If f is a func on 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 a ∆x→0 i=1
  235. 235. Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1
  236. 236. Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S)
  237. 237. Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) f(x) — integrand
  238. 238. Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integra on (a is the lower limit and b the upper limit)
  239. 239. Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integra on (a is the lower limit and b the upper limit) dx — ??? (a parenthesis? an infinitesimal? a variable?)
  240. 240. Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integra on (a is the lower limit and b the upper limit) dx — ??? (a parenthesis? an infinitesimal? a variable?) The process of compu ng an integral is called integra on or quadrature
  241. 241. The limit can be simplified Theorem If f is con nuous on [a, b] or if f has only finitely many jump discon nui es, then f is integrable on [a, b]; that is, the definite ∫ b integral f(x) dx exists. a

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