The document discusses the definite integral, including computing it using Riemann sums, estimating it using approximations like the midpoint rule, and reasoning about its properties. It outlines the topics to be covered, such as recalling previous concepts and comparing properties of integrals. Formulas are provided for calculating Riemann sums using different representative points within the intervals.

1. Section 5.2
The Definite Integral
V63.0121.002.2010Su, Calculus I
New York University
June 17, 2010
Announcements
. . . . . .

2. Announcements
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 2 / 32

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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