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- 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. 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. 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. 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. 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. 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. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. . b
- 8. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. h . b
- 9. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. h .
- 10. Easy Areas: Parallelogram By cu ng and pas ng, a parallelogram can be made into a rectangle. h . b
- 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. Easy Areas: Triangle By copying and pas ng, a triangle can be made into a parallelogram. . b
- 13. Easy Areas: Triangle By copying and pas ng, a triangle can be made into a parallelogram. h . b
- 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. Easy Areas: Other Polygons Any polygon can be triangulated, so its area can be found by summing the areas of the triangles: . .
- 16. Hard Areas: Curved Regions . ???
- 17. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (a er Euclid) Geometer Weapons engineer
- 18. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (a er Euclid) Geometer Weapons engineer
- 19. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (a er Euclid) Geometer Weapons engineer
- 20. Archimedes and the Parabola . Archimedes found areas of a sequence of triangles inscribed in a parabola. A=
- 21. Archimedes and the Parabola 1 . Archimedes found areas of a sequence of triangles inscribed in a parabola. A=1
- 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. 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. 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. Summing the series We need to know the value of the series 1 1 1 1+ + + ··· + n + ··· 4 16 4
- 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. 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. 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. 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. 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. Cavalieri Italian, 1598–1647 Revisited the area problem with a diﬀerent perspec ve
- 32. Cavalieri’s method Divide up the interval into pieces and 2 y=x measure the area of the inscribed rectangles: . 0 1
- 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. 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. 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. 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. 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. 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. 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. 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. What is Ln? 1 Divide the interval [0, 1] into n pieces. Then each has width . n
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Forming Riemann sums We have many choices of representa ve points to approximate the area in each subinterval. …even random points! . x
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4)
- 218. Analogies The Tangent Problem The Area Problem (Ch. 5) (Ch. 2–4) Want the slope of a curve
- 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. 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. 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. 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. 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. 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. 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. Distances Just like area = length × width, we have distance = rate × me. So here is another use for Riemann sums.
- 227. Application: Dead Reckoning
- 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. 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. 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. 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. 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. 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. 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. Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1
- 236. Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S)
- 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. 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. 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. 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. 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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