We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties

1. Section 5.2
The Definite Integral
Math 1a
December 7, 2007
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2. Outline
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral

3. 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
n
b
f (x) dx = lim f (ci ) ∆x
∆x→0
a i=1

4. Notation/Terminology
b
f (x) dx
a

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

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

7. 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)

8. 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?)

9. 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

10. 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

11. 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
∆x = and xi = a + i ∆x
n

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

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

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

15. Example
1
4
Estimate dx using the midpoint rule and four divisions.
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)

16. Example
1
4
Estimate dx using the midpoint rule and four divisions.
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)
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64

17. Example
1
4
Estimate dx using the midpoint rule and four divisions.
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)
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64
150, 166, 784
≈ 3.1468
=
47, 720, 465

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

19. 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

20. 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
2. [f (x) + g (x)] dx = f (x) dx + g (x) dx.
a a a

21. 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
2. [f (x) + g (x)] dx = f (x) dx + g (x) dx.
a a a
b b
3. cf (x) dx = c f (x) dx.
a a

22. 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
2. [f (x) + g (x)] dx = f (x) dx + g (x) dx.
a a a
b b
3. cf (x) dx = c f (x) dx.
a a
b b b
[f (x) − g (x)] dx = f (x) dx −
4. g (x) dx.
a a a

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

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

25. 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
5. f (x) dx = f (x) dx + f (x) dx for all a, b, and c.
a a b

26. 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
(b) f (x) dx.
4

27. 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

28. 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

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

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

31. 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

32. 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

33. 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

34. Example
2
1
Estimate dx using the comparison properties.
x
1

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