The limit of Riemann Sums has a name: the definite integral. We compute a few "easy" ones and show general properties.

1. Section 5.2
The Definite Integral
V63.0121, Calculus I
April 16, 2009
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. . . . . .

2. Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .

3. 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.
b−a
. For each i between 1 and n, let xi be the ith
Then ∆x =
n
step between 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
xn = a + n · =b
n
.. . .. x
.
. 0 . 1 . . . . i . . .. n−1. n
x x. x. x x
. . . . . .

4. 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
.
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
. . . . . .

5. 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
.
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
. . . . . .

6. 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
.
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
. . . . . .

7. Forming Riemann sums
We have many choices of representative points to approximate
the area in each subinterval.
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
. . . . . .

8. 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
. x
.
matter what choice of ci we
made.
. . . . . .

9. 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
. x
.
matter what choice of ci we
made.
. . . . . .

10. 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
. . . x
.
matter what choice of ci we
made.
. . . . . .

11. 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
. . . . x
.
matter what choice of ci we
made.
. . . . . .

12. 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
. . . . . x
.
matter what choice of ci we
made.
. . . . . .

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
. . . . . . x
.
matter what choice of ci we
made.
. . . . . .

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
....... x
.
matter what choice of ci we
made.
. . . . . .

15. 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
........ x
.
matter what choice of ci we
made.
. . . . . .

16. 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
......... x
.
matter what choice of ci we
made.
. . . . . .

17. 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
.......... x
.
matter what choice of ci we
made.
. . . . . .

18. 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
........... x
.
matter what choice of ci we
made.
. . . . . .

19. 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
............ x
.
matter what choice of ci we
made.
. . . . . .

20. 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
............. x
.
matter what choice of ci we
made.
. . . . . .

21. 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
.............. x
.
matter what choice of ci we
made.
. . . . . .

22. 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
............... x
.
matter what choice of ci we
made.
. . . . . .

23. 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
................ x
.
matter what choice of ci we
made.
. . . . . .

24. 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
................. x
.
matter what choice of ci we
made.
. . . . . .

25. 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
.................. x
.
matter what choice of ci we
made.
. . . . . .

26. 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
................... x
.
matter what choice of ci we
made.
. . . . . .

27. 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
.................... x
.
matter what choice of ci we
made.
. . . . . .

28. 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
.....................
. x
.
matter what choice of ci we
made.
. . . . . .

29. 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
......................
. x
.
matter what choice of ci we
made.
. . . . . .

30. 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
.......................
. x
.
matter what choice of ci we
made.
. . . . . .

31. 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
........................
. x
.
matter what choice of ci we
made.
. . . . . .

32. 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
.........................
. x
.
matter what choice of ci we
made.
. . . . . .

33. 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
..........................
. x
.
matter what choice of ci we
made.
. . . . . .

34. 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
...........................
. x
.
matter what choice of ci we
made.
. . . . . .

35. 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
............................
. x
.
matter what choice of ci we
made.
. . . . . .

36. 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
.............................
. x
.
matter what choice of ci we
made.
. . . . . .

37. 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
..............................
. x
.
matter what choice of ci we
made.
. . . . . .

38. Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .

39. The definite integral as a limit
Definition
If f is a function 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
∆x→0
a i=1
. . . . . .

40. Notation/Terminology
∫ b
f(x) dx
a
. . . . . .

41. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
. . . . . .

42. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
. . . . . .

43. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
. . . . . .

44. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
. . . . . .

45. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
The process of computing an integral is called integration or
quadrature
. . . . . .

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

47. The limit can be simplified
Theorem
If f is continuous on [a, b] or if f has only finitely many jump
discontinuities, then f is integrable on [a, b]; that is, the definite
∫b
integral f(x) dx exists.
a
Theorem
If f is integrable on [a, b] then
∫ n
∑
b
f(x) dx = lim f(xi )∆x,
n→∞
a i=1
where
b−a
and xi = a + i ∆x
∆x =
n
. . . . . .

48. Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .

49. Estimating the Definite Integral
Given a partition of [a, b] into n pieces, let ¯i be the midpoint of
x
[xi−1 , xi ]. Define
n
∑
Mn = f(¯i ) ∆x.
x
i=1
. . . . . .

50. Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
. . . . . .

51. Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
2 2 2 1 + (7/8)2
4 1 + (1/8) 1 + (3/8) 1 + (5/8)
. . . . . .

52. Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2
( )
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64
. . . . . .

53. Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2
( )
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64
150, 166, 784
≈ 3.1468
=
47, 720, 465
. . . . . .

54. Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .

55. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫b
c dx = c(b − a)
1.
a
. . . . . .

56. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫b
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
. . . . . .

57. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫b
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
∫ ∫
b b
cf(x) dx = c f(x) dx.
3.
a a
. . . . . .

58. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫b
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
∫ ∫
b b
cf(x) dx = c f(x) dx.
3.
a a
∫ ∫ ∫
b b b
[f(x) − g(x)] dx = f(x) dx − g(x) dx.
4.
a a a
. . . . . .

59. More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
. . . . . .

60. More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
∫ a
f(x) dx = 0
a
. . . . . .

61. More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
∫ a
f(x) dx = 0
a
This allows us to have
∫c ∫b ∫ c
f(x) dx = f(x) dx + f(x) dx for all a, b, and c.
5.
a a b
. . . . . .

62. Example
Suppose f and g are functions with
∫4
f(x) dx = 4
0
∫5
f(x) dx = 7
0
∫5
g(x) dx = 3.
0
Find
∫5
[2f(x) − g(x)] dx
(a)
0
∫5
f(x) dx.
(b)
4
. . . . . .

63. Solution
We have
(a)
∫ ∫ ∫
5 5 5
[2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx
0 0 0
= 2 · 7 − 3 = 11
. . . . . .

64. Solution
We have
(a)
∫ ∫ ∫
5 5 5
[2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx
0 0 0
= 2 · 7 − 3 = 11
(b)
∫ ∫ ∫
5 5 4
f(x) dx −
f(x) dx = f(x) dx
4 0 0
=7−4=3
. . . . . .

65. Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .

66. Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
. . . . . .

67. Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f(x) ≥ 0 for all x in [a, b], then
∫ b
f(x) dx ≥ 0
a
. . . . . .

68. Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f(x) ≥ 0 for all x in [a, b], then
∫ b
f(x) dx ≥ 0
a
7. If f(x) ≥ g(x) for all x in [a, b], then
∫ ∫
b b
f(x) dx ≥ g(x) dx
a a
. . . . . .

69. Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f(x) ≥ 0 for all x in [a, b], then
∫ b
f(x) dx ≥ 0
a
7. If f(x) ≥ g(x) for all x in [a, b], then
∫ ∫
b b
f(x) dx ≥ g(x) dx
a a
8. If m ≤ f(x) ≤ M for all x in [a, b], then
∫ b
m(b − a) ≤ f(x) dx ≤ M(b − a)
a
. . . . . .

70. Example
∫ 2
1
dx using the comparison properties.
Estimate
x
1
. . . . . .

71. Example
∫ 2
1
dx using the comparison properties.
Estimate
x
1
Solution
Since
1 1
≤x≤
2 1
for all x in [1, 2], we have
∫ 2
1 1
·1≤ dx ≤ 1 · 1
x
2 1
. . . . . .