We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.

1. Sec on 5.1–5.2
Areas and Distances, The Definite
Integral
V63.0121.011: Calculus I
Professor Ma hew Leingang
New York University
April 25, 2011
.

2. Announcements
Quiz 5 on Sec ons
4.1–4.4 April 28/29
Final Exam Thursday May
12, 2:00–3:50pm
cumula ve
loca on TBD
old exams on common
website

3. Objectives from Section 5.1
Compute the area of a region by
approxima ng it with rectangles
and le ng the size of the
rectangles tend to zero.
Compute the total distance
traveled by a par cle by
approxima ng it as distance =
(rate)( me) and le ng the me
intervals over which one
approximates tend to zero.

4. Objectives from Section 5.2
Compute the definite integral
using a limit of Riemann sums
Es mate the definite integral
using a Riemann sum (e.g.,
Midpoint Rule)
Reason with the definite integral
using its elementary proper es.

5. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applica ons
The definite integral as a limit
Es ma ng the Definite Integral
Proper es of the integral
Comparison Proper es of the Integral

6. Easy Areas: Rectangle
Defini on
The area of a rectangle with dimensions ℓ and w is the product
A = ℓw.
w
.
ℓ
It may seem strange that this is a defini on and not a theorem but
we have to start somewhere.

7. Easy Areas: Parallelogram
By cu ng and pas ng, a parallelogram can be made into a rectangle.
.
b

8. Easy Areas: Parallelogram
By cu ng and pas ng, a parallelogram can be made into a rectangle.
h
.
b

9. Easy Areas: Parallelogram
By cu ng and pas ng, a parallelogram can be made into a rectangle.
h
.

10. Easy Areas: Parallelogram
By cu ng and pas ng, a parallelogram can be made into a rectangle.
h
.
b

11. Easy Areas: Parallelogram
By cu ng and pas ng, a parallelogram can be made into a rectangle.
h
.
So b
Fact
The area of a parallelogram of base width b and height h is
A = bh

12. Easy Areas: Triangle
By copying and pas ng, a triangle can be made into a parallelogram.
.
b

13. Easy Areas: Triangle
By copying and pas ng, a triangle can be made into a parallelogram.
h
.
b

14. Easy Areas: Triangle
By copying and pas ng, a triangle can be made into a parallelogram.
h
.
So b
Fact
The area of a triangle of base width b and height h is
1
A = bh
2

15. Easy Areas: Other Polygons
Any polygon can be triangulated, so its area can be found by
summing the areas of the triangles:
.
.

16. Hard Areas: Curved Regions
.
???

17. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC
– 212 BC (a er Euclid)
Geometer
Weapons engineer

18. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC
– 212 BC (a er Euclid)
Geometer
Weapons engineer

19. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC
– 212 BC (a er Euclid)
Geometer
Weapons engineer

20. Archimedes and the Parabola
.
Archimedes found areas of a sequence of triangles inscribed in a
parabola.
A=

21. Archimedes and the Parabola
1
.
Archimedes found areas of a sequence of triangles inscribed in a
parabola.
A=1

22. Archimedes and the Parabola
1
1 1
8 8
.
Archimedes found areas of a sequence of triangles inscribed in a
parabola.
1
A=1+2·
8

23. Archimedes and the Parabola
1 1
64 64
1
1 1
8 8
1 1
64 64
.
Archimedes found areas of a sequence of triangles inscribed in a
parabola.
1 1
A=1+2· +4· + ···
8 64

24. Archimedes and the Parabola
1 1
64 64
1
1 1
8 8
1 1
64 64
.
Archimedes found areas of a sequence of triangles inscribed in a
parabola.
1 1 1 1 1
A=1+2· +4· + ··· = 1 + + + ··· + n + ···
8 64 4 16 4

25. Summing the series
We need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4

26. Summing a geometric series
Fact
For any number r and any posi ve integer n,
(1 − r)(1 + r + r2 + · · · + rn ) = 1 − rn+1 .

27. Summing a geometric series
Fact
For any number r and any posi ve integer n,
(1 − r)(1 + r + r2 + · · · + rn ) = 1 − rn+1 .
Proof.
(1 − r)(1 + r + r2 + · · · + rn )
= (1 + r + r2 + · · · + rn ) − r(1 + r + r2 + · · · + rn )
= (1 + r + r2 + · · · + rn ) − (r + r2 + r3 · · · + rn + rn+1 )
= 1 − rn+1

28. Summing a geometric series
Fact
For any number r and any posi ve integer n,
(1 − r)(1 + r + r2 + · · · + rn ) = 1 − rn+1 .
Corollary
1 − rn+1
1 + r + ··· + r =n
1−r

29. Summing the series
We need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
Using the corollary,
1 1 1 1 − (1/4)n+1
1+ + + ··· + n =
4 16 4 1 − 1/4

30. Summing the series
We need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
Using the corollary,
1 1 1 1 − (1/4)n+1 1 4
1+ + + ··· + n = → 3 = as n → ∞.
4 16 4 1 − 1/4 /4 3

31. Cavalieri
Italian,
1598–1647
Revisited
the area
problem
with a
different
perspec ve

32. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
.
0 1

33. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
1
L2 =
8
.
0 1 1
2

34. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
1
L2 =
8
L3 =
.
0 1 2 1
3 3

35. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
.
0 1 2 1
3 3

36. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
L4 =
.
0 1 2 3 1
4 4 4

37. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
. 64 64 64 64
0 1 2 3 1
4 4 4

38. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
. 64 64 64 64
0 1 2 3 4 1 L5 =
5 5 5 5

39. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
. 64 64 64 64
1 4 9 16 30
0 1 2 3 4 1 L5 = + + + =
125 125 125 125 125
5 5 5 5

40. Cavalieri’s method
Divide up the interval into pieces and
2
y=x measure the area of the inscribed
rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
. 64 64 64 64
1 4 9 16 30
0 1 L5 = + + + =
125 125 125 125 125
Ln =?

41. What is Ln? 1
Divide the interval [0, 1] into n pieces. Then each has width .
n

42. What is Ln? 1
Divide the interval [0, 1] into n pieces. Then each has width . The
n
rectangle over the ith interval and under the parabola has area
( )2
1 i−1 (i − 1)2
· = .
n n n3

43. What is Ln? 1
Divide the interval [0, 1] into n pieces. Then each has width . The
n
rectangle over the ith interval and under the parabola has area
( )2
1 i−1 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = 3 + 3 + · · · + =
n n n3 n3

44. The Square Pyramidial Numbers
Fact
Let n be a posi ve integer. Then
n(n − 1)(2n − 1)
1 + 22 + 32 + · · · + (n − 1)2 =
6
This formula was known to the Arabs and discussed by Fibonacci in
his book Liber Abaci.

45. What is Ln? 1
Divide the interval [0, 1] into n pieces. Then each has width . The
n
rectangle over the ith interval and under the parabola has area
( )2
1 i−1 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = 3 + 3 + · · · + =
n n n3 n3
So
n(n − 1)(2n − 1)
Ln =
6n3

46. What is Ln? 1
Divide the interval [0, 1] into n pieces. Then each has width . The
n
rectangle over the ith interval and under the parabola has area
( )2
1 i−1 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = 3 + 3 + · · · + =
n n n3 n3
So
n(n − 1)(2n − 1) 1
Ln = →
6n3 3
as n → ∞.

47. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n

48. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3

49. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
1 + 23 + 33 + · · · + (n − 1)3
=
n4

50. Nicomachus’s Theorem
Fact (Nicomachus 1st c. CE, Aryabhata 5th c., Al-Karaji 11th c.)
1 + 23 + 33 + · · · + (n − 1)3 = [1 + 2 + · · · + (n − 1)]2
[1 ]2
= 2 n(n − 1)

51. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
1 + 23 + 33 + · · · + (n − 1)3
=
n4
n2 (n − 1)2
=
4n4

52. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
1 + 23 + 33 + · · · + (n − 1)3
=
n4
n2 (n − 1)2 1
= →
4n4 4
as n → ∞.

53. Cavalieri’s method with different heights
1 13 1 23 1 n3
Rn = · 3 + · 3 + ··· + · 3
n n n n n n
1 + 2 + 3 + ··· + n
3 3 3 3
=
n4
1 [ ]2
= 4 1 n(n + 1)
n 2
n2 (n + 1)2 1
. = →
4n4 4
as n → ∞.

54. Cavalieri’s method with different heights
1 13 1 23 1 n3
Rn = · 3 + · 3 + ··· + · 3
n n n n n n
1 + 2 + 3 + ··· + n
3 3 3 3
=
n4
1 [ ]2
= 4 1 n(n + 1)
n 2
n2 (n + 1)2 1
. = →
4n4 4
as n → ∞.
So even though the rectangles overlap, we s ll get the same answer.

55. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applica ons
The definite integral as a limit
Es ma ng the Definite Integral
Proper es of the integral
Comparison Proper es of the Integral

56. Cavalieri’s method in general
Problem
Let f be a posi ve func on defined
on the interval [a, b]. Find the
area between x = a, x = b, y = 0,
and y = f(x).
.
. x x
x0 x1. . . xi . . xn−1 n

57. Cavalieri’s method in general
For each posi ve integer n, divide up the interval into n pieces. Then
b−a
∆x = . For each i between 1 and n, let xi be the ith step
n
between a and b.
x0 = a
b−a
x1 = x0 + ∆x = a +
n
b−a
x2 = x1 + ∆x = a + 2 · ...
n
b−a
xi = a + i · ...
n
. b−a
. x x
x0 x1. . . xi . . xn−1 n xn = a + n · =b
n

58. Forming Riemann Sums
Choose ci to be a point in the ith interval [xi−1 , xi ]. Form the
Riemann sum
∑
n
Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x
i=1
Thus we approximate area under a curve by a sum of areas of
rectangles.

59. Forming Riemann sums
We have many choices of representa ve points to approximate the
area in each subinterval.
le endpoints…
∑
n
Ln = f(xi−1 )∆x
i=1
. x

60. Forming Riemann sums
We have many choices of representa ve points to approximate the
area in each subinterval.
right endpoints…
∑
n
Rn = f(xi )∆x
i=1
. x

61. Forming Riemann sums
We have many choices of representa ve points to approximate the
area in each subinterval.
midpoints…
∑ ( xi−1 + xi )
n
Mn = f ∆x
i=1
2
. x

62. Forming Riemann sums
We have many choices of representa ve points to approximate the
area in each subinterval.
the maximum value on the
interval…
∑
n
Un = max {f(x)} ∆x
xi−1 ≤x≤xi
i=1
. x

63. Forming Riemann sums
We have many choices of representa ve points to approximate the
area in each subinterval.
the minimum value on the
interval…
∑
n
Ln = min {f(x)} ∆x
xi−1 ≤x≤xi
i=1
. x

64. Forming Riemann sums
We have many choices of representa ve points to approximate the
area in each subinterval.
…even random points!
. x

65. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make.

66. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make.

67. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L1 = 3.0
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

68. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L2 = 5.25
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

69. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L3 = 6.0
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

70. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L4 = 6.375
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

71. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L5 = 6.59988
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

72. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L6 = 6.75
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

73. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L7 = 6.85692
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

74. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L8 = 6.9375
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

75. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L9 = 6.99985
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

76. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L10 = 7.04958
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

77. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L11 = 7.09064
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

78. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L12 = 7.125
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

79. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L13 = 7.15332
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

80. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L14 = 7.17819
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

81. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L15 = 7.19977
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

82. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L16 = 7.21875
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

83. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L17 = 7.23508
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

84. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L18 = 7.24927
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

85. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L19 = 7.26228
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

86. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L20 = 7.27443
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

87. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L21 = 7.28532
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

88. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L22 = 7.29448
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

89. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L23 = 7.30406
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

90. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L24 = 7.3125
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

91. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L25 = 7.31944
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

92. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L26 = 7.32559
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

93. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L27 = 7.33199
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

94. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L28 = 7.33798
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

95. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L29 = 7.34372
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

96. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L30 = 7.34882
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. le endpoints

97. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R1 = 12.0
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

98. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R2 = 9.75
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

99. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R3 = 9.0
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

100. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R4 = 8.625
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

101. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R5 = 8.39969
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

102. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R6 = 8.25
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

103. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R7 = 8.14236
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

104. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R8 = 8.0625
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

105. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R9 = 7.99974
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

106. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R10 = 7.94933
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

107. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R11 = 7.90868
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

108. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R12 = 7.875
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

109. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R13 = 7.84541
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

110. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R14 = 7.8209
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

111. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R15 = 7.7997
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

112. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R16 = 7.78125
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

113. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R17 = 7.76443
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

114. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R18 = 7.74907
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

115. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R19 = 7.73572
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

116. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R20 = 7.7243
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

117. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R21 = 7.7138
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

118. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R22 = 7.70335
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

119. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R23 = 7.69531
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

120. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R24 = 7.6875
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

121. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R25 = 7.67934
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

122. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R26 = 7.6715
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

123. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R27 = 7.66508
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

124. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R28 = 7.6592
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

125. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R29 = 7.65388
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

126. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump R30 = 7.64864
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. right endpoints

127. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M1 = 7.5
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

128. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M2 = 7.5
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

129. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M3 = 7.5
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

130. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M4 = 7.5
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

131. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M5 = 7.4998
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

132. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M6 = 7.5
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

133. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M7 = 7.4996
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

134. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M8 = 7.5
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

135. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M9 = 7.49977
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

136. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M10 = 7.49947
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

137. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M11 = 7.49966
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

138. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M12 = 7.5
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

139. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M13 = 7.49937
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

140. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M14 = 7.49954
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

141. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M15 = 7.49968
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

142. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M16 = 7.49988
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

143. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M17 = 7.49974
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

144. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M18 = 7.49916
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

145. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M19 = 7.49898
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

146. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M20 = 7.4994
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

147. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M21 = 7.49951
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

148. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M22 = 7.49889
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

149. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M23 = 7.49962
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

150. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M24 = 7.5
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

151. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M25 = 7.49939
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

152. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M26 = 7.49847
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

153. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M27 = 7.4985
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

154. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M28 = 7.4986
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

155. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M29 = 7.49878
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

156. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump M30 = 7.49872
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. midpoints

157. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U1 = 12.0
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

158. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U2 = 10.55685
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

159. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U3 = 10.0379
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

160. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U4 = 9.41515
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

161. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U5 = 8.96004
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

162. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U6 = 8.76895
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

163. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U7 = 8.6033
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

164. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U8 = 8.45757
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

165. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U9 = 8.34564
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

166. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U10 = 8.27084
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

167. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U11 = 8.20132
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

168. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U12 = 8.13838
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

169. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U13 = 8.0916
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

170. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U14 = 8.05139
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

171. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U15 = 8.01364
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

172. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U16 = 7.98056
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

173. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U17 = 7.9539
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

174. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U18 = 7.92815
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

175. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U19 = 7.90414
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

176. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U20 = 7.88504
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

177. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U21 = 7.86737
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

178. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U22 = 7.84958
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

179. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U23 = 7.83463
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

180. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U24 = 7.82187
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

181. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U25 = 7.80824
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

182. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U26 = 7.79504
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

183. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U27 = 7.78429
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

184. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U28 = 7.77443
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

185. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U29 = 7.76495
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

186. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump U30 = 7.7558
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. maximum points

187. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L1 = 3.0
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

188. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L2 = 4.44312
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

189. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L3 = 4.96208
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

190. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L4 = 5.58484
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

191. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L5 = 6.0395
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

192. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L6 = 6.23103
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

193. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L7 = 6.39577
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

194. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L8 = 6.54242
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

195. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L9 = 6.65381
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

196. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L10 = 6.72797
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

197. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L11 = 6.7979
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

198. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L12 = 6.8616
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

199. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L13 = 6.90704
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

200. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L14 = 6.94762
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

201. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L15 = 6.98575
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

202. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L16 = 7.01942
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

203. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L17 = 7.04536
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

204. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L18 = 7.07005
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

205. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L19 = 7.09364
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

206. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L20 = 7.1136
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

207. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L21 = 7.13155
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

208. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L22 = 7.14804
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

209. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L23 = 7.16441
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

210. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L24 = 7.17812
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

211. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L25 = 7.19025
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

212. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L26 = 7.2019
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

213. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L27 = 7.21265
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

214. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L28 = 7.22269
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

215. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L29 = 7.23251
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

216. Theorem of the Day
Theorem
If f is a con nuous func on on
[a, b] or has finitely many jump L30 = 7.24162
discon nui es, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no . x
ma er what choice of ci we make. minimum points

217. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4)

218. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4)
Want the slope of a curve

219. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region

220. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of
lines

221. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of Only know the area of
lines polygons

222. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of Only know the area of
lines polygons
Approximate curve with a
line

223. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of Only know the area of
lines polygons
Approximate curve with a Approximate region with
line polygons

224. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of Only know the area of
lines polygons
Approximate curve with a Approximate region with
line polygons
Take limit over be er and Take limit over be er and
be er approxima ons be er approxima ons

225. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applica ons
The definite integral as a limit
Es ma ng the Definite Integral
Proper es of the integral
Comparison Proper es of the Integral

226. Distances
Just like area = length × width, we have
distance = rate × me.
So here is another use for Riemann sums.

227. Application: Dead Reckoning

228. Computing position by Dead Reckoning
Example
A sailing ship is cruising back and forth along a channel (in a straight
line). At noon the ship’s posi on and velocity are recorded, but
shortly therea er a storm blows in and posi on is impossible to
measure. The velocity con nues to be recorded at thirty-minute
intervals.

229. Computing position by Dead Reckoning
Example
Time 12:00 12:30 1:00 1:30 2:00
Speed (knots) 4 8 12 6 4
Direc on E E E E W
Time 2:30 3:00 3:30 4:00
Speed 3 3 5 9
Direc on W E E E
Es mate the ship’s posi on at 4:00pm.

230. Solution
Solu on
We es mate that the speed of 4 knots (nau cal miles per hour) is
maintained from 12:00 un l 12:30. So over this me interval the
ship travels ( )( )
4 nmi 1
hr = 2 nmi
hr 2
We can con nue for each addi onal half hour and get
distance = 4 × 1/2 + 8 × 1/2 + 12 × 1/2
+ 6 × 1/2 − 4 × 1/2 − 3 × 1/2 + 3 × 1/2 + 5 × 1/2 = 15.5
So the ship is 15.5 nmi east of its original posi on.

231. Analysis
This method of measuring posi on by recording velocity was
necessary un l global-posi oning satellite technology became
widespread
If we had velocity es mates at finer intervals, we’d get be er
es mates.
If we had velocity at every instant, a limit would tell us our
exact posi on rela ve to the last me we measured it.

232. Other uses of Riemann sums
Anything with a product!
Area, volume
Anything with a density: Popula on, mass
Anything with a “speed:” distance, throughput, power
Consumer surplus
Expected value of a random variable

233. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applica ons
The definite integral as a limit
Es ma ng the Definite Integral
Proper es of the integral
Comparison Proper es of the Integral

234. The definite integral as a limit
Defini on
If f is a func on defined on [a, b], the definite integral of f from a to
b is the number
∫ b ∑n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1

235. Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1

236. Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)

237. Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
f(x) — integrand

238. Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integra on (a is the lower limit and b the
upper limit)

239. Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integra on (a is the lower limit and b the
upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)

240. Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integra on (a is the lower limit and b the
upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
The process of compu ng an integral is called integra on or
quadrature

241. The limit can be simplified
Theorem
If f is con nuous on [a, b] or if f has only finitely many jump
discon nui es, then f is integrable on [a, b]; that is, the definite
∫ b
integral f(x) dx exists.
a

242. The limit can be simplified
Theorem
If f is con nuous on [a, b] or if f has only finitely many jump
discon nui es, then f is integrable on [a, b]; that is, the definite
∫ b
integral f(x) dx exists.
a
So we can find the integral by compu ng the limit of any sequence
of Riemann sums that we like,

243. Example
∫ 3
Find x dx
0

244. Example
∫ 3
Find x dx
0
Solu on
3 3i
For any n we have ∆x = and for each i between 0 and n, xi = .
n n

245. Example
∫ 3
Find x dx
0
Solu on
3 3i
For any n we have ∆x = and for each i between 0 and n, xi = .
n n
For each i, take xi to represent the func on on the ith interval.

246. Example
∫ 3
Find x dx
0
Solu on
3 3i
For any n we have ∆x = and for each i between 0 and n, xi = .
n n
For each i, take xi to represent the func on on the ith interval. So
∫ 3
x dx = lim Rn
0 n→∞

247. Example
∫ 3
Find x dx
0
Solu on
3 3i
For any n we have ∆x = and for each i between 0 and n, xi = .
n n
For each i, take xi to represent the func on on the ith interval. So
∫ 3 ∑
n
x dx = lim Rn = lim f(xi ) ∆x
0 n→∞ n→∞
i=1

248. Example
∫ 3
Find x dx
0
Solu on
3 3i
For any n we have ∆x = and for each i between 0 and n, xi = .
n n
For each i, take xi to represent the func on on the ith interval. So
∫ 3 ∑n ∑ ( 3i ) ( 3 )
n
x dx = lim Rn = lim f(xi ) ∆x = lim
0 n→∞ n→∞
i=1
n→∞
i=1
n n