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# Lesson 25: The Definite Integral

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• @Matthew: Thanks a lot! That's a really nice site. I'll definitely start hanging out there. I found this page via a link you provided on another page while looking for help with the \pgfmathsetmacro command; made last night's tasks much less strenuous.

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• @Jrock4real: thanks for the compliments. I've been doing beamer slideshows for my classes for about five years. The Calculus I slides are the ones into which I've put the most work—I'm able to improve them continually from term to term. So your presentations will get gradually better as you accumulate experience and your body of work.

I've been hanging out a lot at http://tex.stackexchange.com/ lately. There are many tikz and beamer users there so feel free to join us!

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### Lesson 25: The Definite Integral

1. 1. Section 5.2 The Definite Integral V63.0121.002.2010Su, Calculus I New York University June 17, 2010 Announcements . . . . . .
2. 2. Announcements . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 2 / 32
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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 3 / 32
4. 4. Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 4 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 5 / 32
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 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
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 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
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 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
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 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
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 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
11. 11. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. …even random points! . . . . . . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
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 . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32