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# Lesson 23: The Definite Integral (slides)

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### Lesson 23: The Definite Integral (slides)

1. 1. Section 5.2 The Definite Integral V63.0121.006/016, Calculus I New York University April 15, 2010 Announcements April 16: Quiz 4 on §§4.1–4.4 April 29: Movie Day!! April 30: Quiz 5 on §§5.1–5.4 Monday, May 10, 12:00noon (not 10:00am as previously announced) Final Exam . . . . . .
2. 2. Announcements April 16: Quiz 4 on §§4.1–4.4 April 29: Movie Day!! April 30: Quiz 5 on §§5.1–5.4 Monday, May 10, 12:00noon (not 10:00am as previously announced) Final Exam . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 2 / 28
3. 3. Objectives Compute the definite integral using a limit of Riemann sums Estimate the definite integral using a Riemann sum (e.g., Midpoint Rule) Reason with the definite integral using its elementary properties. . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 3 / 28
4. 4. Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 4 / 28
5. 5. Cavalieri's method in general Let f be a positive function defined on the interval [a, b]. We want to find the area between x = a, x = b, y = 0, and y = f(x). For each positive integer n, divide up the interval into n pieces. Then b−a ∆x = . For each i between 1 and n, let xi be the ith step between n a and b. So x0 = a b−a x1 = x0 + ∆x = a + n b−a x2 = x1 + ∆x = a + 2 · ... n b−a xi = a + i · ... n b−a . . . . . x . xn = a + n · =b . 0 . 1 . . . . i . . .xn−1. n x x x x n . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 5 / 28
6. 6. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. left endpoints… ∑ n Ln = f(xi−1 )∆x i=1 . . . . . . . x . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
7. 7. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. right endpoints… ∑ n Rn = f(xi )∆x i=1 . . . . . . . x . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
8. 8. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. midpoints… ∑ ( xi−1 + xi ) n Mn = f ∆x 2 i=1 . . . . . . . x . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
9. 9. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. the minimum value on the interval… . . . . . . . x . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
10. 10. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. the maximum value on the interval… . . . . . . . x . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
11. 11. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. …even random points! . . . . . . . x . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
12. 12. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. …even random points! . . . . . . . . x In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum ∑ n Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 6 / 28
13. 13. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no matter what choice of ci we make. . x . . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
14. 14. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no matter what choice of ci we make. . x . . . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
15. 15. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 1 = 3.0 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
16. 16. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 2 = 5.25 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
17. 17. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 3 = 6.0 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
18. 18. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 4 = 6.375 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
19. 19. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 5 = 6.59988 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
20. 20. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 6 = 6.75 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
21. 21. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 7 = 6.85692 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
22. 22. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 8 = 6.9375 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
23. 23. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 9 = 6.99985 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
24. 24. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 10 = 7.04958 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
25. 25. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 11 = 7.09064 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
26. 26. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 12 = 7.125 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
27. 27. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 13 = 7.15332 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
28. 28. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 14 = 7.17819 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
29. 29. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 7.19977 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
30. 30. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 16 = 7.21875 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
31. 31. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 17 = 7.23508 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
32. 32. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 18 = 7.24927 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
33. 33. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 19 = 7.26228 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
34. 34. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 20 = 7.27443 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
35. 35. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 21 = 7.28532 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
36. 36. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 22 = 7.29448 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
37. 37. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 23 = 7.30406 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
38. 38. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 24 = 7.3125 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
39. 39. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 25 = 7.31944 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
40. 40. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 26 = 7.32559 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
41. 41. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 27 = 7.33199 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
42. 42. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 28 = 7.33798 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
43. 43. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 29 = 7.34372 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
44. 44. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 30 = 7.34882 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
45. 45. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 1 = 12.0 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
46. 46. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 2 = 9.75 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
47. 47. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 3 = 9.0 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
48. 48. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 4 = 8.625 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
49. 49. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 5 = 8.39969 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
50. 50. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 6 = 8.25 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
51. 51. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 7 = 8.14236 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
52. 52. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 8 = 8.0625 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
53. 53. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 9 = 7.99974 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
54. 54. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 10 = 7.94933 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
55. 55. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 11 = 7.90868 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
56. 56. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 12 = 7.875 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
57. 57. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 13 = 7.84541 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
58. 58. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 14 = 7.8209 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
59. 59. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 7.7997 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
60. 60. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 16 = 7.78125 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
61. 61. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 17 = 7.76443 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
62. 62. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 18 = 7.74907 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
63. 63. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 19 = 7.73572 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
64. 64. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 20 = 7.7243 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
65. 65. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 21 = 7.7138 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
66. 66. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 22 = 7.70335 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
67. 67. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 23 = 7.69531 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
68. 68. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 24 = 7.6875 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
69. 69. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 25 = 7.67934 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
70. 70. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 26 = 7.6715 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
71. 71. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 27 = 7.66508 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
72. 72. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 28 = 7.6592 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
73. 73. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 29 = 7.65388 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
74. 74. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 30 = 7.64864 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
75. 75. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 1 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
76. 76. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 2 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
77. 77. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 3 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
78. 78. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 4 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
79. 79. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 5 = 7.4998 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
80. 80. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 6 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
81. 81. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 7 = 7.4996 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
82. 82. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 8 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
83. 83. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 9 = 7.49977 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
84. 84. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 10 = 7.49947 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
85. 85. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 11 = 7.49966 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
86. 86. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 12 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
87. 87. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 13 = 7.49937 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
88. 88. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 14 = 7.49954 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
89. 89. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 7.49968 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28
90. 90. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 16 = 7.49988 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . . V63.0121.006/016, Calculus I (NYU) Section 5.2 The Definite Integral April 15, 2010 7 / 28