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