lesson 30: the definite integral
DESCRIPTION
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative propertiesTRANSCRIPT
Section 5.2The Definite Integral
Math 1a
December 7, 2007
Announcements
I my next office hours: Monday 1–2, Tuesday 3–4 (SC 323)
I MT II is graded. You’ll get it back today
I Final seview sessions: Wed 1/9 and Thu 1/10 in Hall D, Sun1/13 in Hall C, all 7–8:30pm
I Final tentatively scheduled for January 17
Outline
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
The definite integral as a limit
DefinitionIf f is a function defined on [a, b], the definite integral of f froma to b is the number∫ b
af (x) dx = lim
∆x→0
n∑i=1
f (ci ) ∆x
Notation/Terminology
∫ b
af (x) dx
I
∫— integral sign (swoopy S)
I f (x) — integrand
I a and b — limits of integration (a is the lower limit and bthe upper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)
I The process of computing an integral is called integration
Notation/Terminology
∫ b
af (x) dx
I
∫— integral sign (swoopy S)
I f (x) — integrand
I a and b — limits of integration (a is the lower limit and bthe upper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)
I The process of computing an integral is called integration
Notation/Terminology
∫ b
af (x) dx
I
∫— integral sign (swoopy S)
I f (x) — integrand
I a and b — limits of integration (a is the lower limit and bthe upper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)
I The process of computing an integral is called integration
Notation/Terminology
∫ b
af (x) dx
I
∫— integral sign (swoopy S)
I f (x) — integrand
I a and b — limits of integration (a is the lower limit and bthe upper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)
I The process of computing an integral is called integration
Notation/Terminology
∫ b
af (x) dx
I
∫— integral sign (swoopy S)
I f (x) — integrand
I a and b — limits of integration (a is the lower limit and bthe upper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)
I The process of computing an integral is called integration
Notation/Terminology
∫ b
af (x) dx
I
∫— integral sign (swoopy S)
I f (x) — integrand
I a and b — limits of integration (a is the lower limit and bthe upper limit)
I dx — ??? (a parenthesis? an infinitesimal? a variable?)
I The process of computing an integral is called integration
The limit can be simplified
TheoremIf f is continuous on [a, b] or if f has only finitely many jumpdiscontinuities, then f is integrable on [a, b]; that is, the definite
integral
∫ b
af (x) dx exists.
TheoremIf f is integrable on [a, b] then∫ b
af (x) dx = lim
n→∞
n∑i=1
f (xi )∆x ,
where
∆x =b − a
nand xi = a + i ∆x
The limit can be simplified
TheoremIf f is continuous on [a, b] or if f has only finitely many jumpdiscontinuities, then f is integrable on [a, b]; that is, the definite
integral
∫ b
af (x) dx exists.
TheoremIf f is integrable on [a, b] then∫ b
af (x) dx = lim
n→∞
n∑i=1
f (xi )∆x ,
where
∆x =b − a
nand xi = a + i ∆x
Outline
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
Estimating the Definite Integral
Given a partition of [a, b] into n pieces, let x̄i be the midpoint of[xi−1, xi ]. Define
Mn =n∑
i=1
f (x̄i ) ∆x .
Example
Estimate
∫ 1
0
4
1 + x2dx using the midpoint rule and four divisions.
Solution
The partition is 0 <1
4<
1
2<
3
4< 1, so the estimate is
M4 =1
4
(4
1 + (1/8)2+
4
1 + (3/8)2+
4
1 + (5/8)2+
4
1 + (7/8)2
)
=1
4
(4
65/64+
4
73/64+
4
89/64+
4
113/64
)=
150, 166, 784
47, 720, 465≈ 3.1468
Example
Estimate
∫ 1
0
4
1 + x2dx using the midpoint rule and four divisions.
Solution
The partition is 0 <1
4<
1
2<
3
4< 1, so the estimate is
M4 =1
4
(4
1 + (1/8)2+
4
1 + (3/8)2+
4
1 + (5/8)2+
4
1 + (7/8)2
)
=1
4
(4
65/64+
4
73/64+
4
89/64+
4
113/64
)=
150, 166, 784
47, 720, 465≈ 3.1468
Example
Estimate
∫ 1
0
4
1 + x2dx using the midpoint rule and four divisions.
Solution
The partition is 0 <1
4<
1
2<
3
4< 1, so the estimate is
M4 =1
4
(4
1 + (1/8)2+
4
1 + (3/8)2+
4
1 + (5/8)2+
4
1 + (7/8)2
)=
1
4
(4
65/64+
4
73/64+
4
89/64+
4
113/64
)
=150, 166, 784
47, 720, 465≈ 3.1468
Example
Estimate
∫ 1
0
4
1 + x2dx using the midpoint rule and four divisions.
Solution
The partition is 0 <1
4<
1
2<
3
4< 1, so the estimate is
M4 =1
4
(4
1 + (1/8)2+
4
1 + (3/8)2+
4
1 + (5/8)2+
4
1 + (7/8)2
)=
1
4
(4
65/64+
4
73/64+
4
89/64+
4
113/64
)=
150, 166, 784
47, 720, 465≈ 3.1468
Outline
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
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
1.
∫ b
ac dx = c(b − a)
2.
∫ b
a[f (x) + g(x)] dx =
∫ b
af (x) dx +
∫ b
ag(x) dx.
3.
∫ b
acf (x) dx = c
∫ b
af (x) dx.
4.
∫ b
a[f (x)− g(x)] dx =
∫ b
af (x) dx −
∫ b
ag(x) dx.
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
1.
∫ b
ac dx = c(b − a)
2.
∫ b
a[f (x) + g(x)] dx =
∫ b
af (x) dx +
∫ b
ag(x) dx.
3.
∫ b
acf (x) dx = c
∫ b
af (x) dx.
4.
∫ b
a[f (x)− g(x)] dx =
∫ b
af (x) dx −
∫ b
ag(x) dx.
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
1.
∫ b
ac dx = c(b − a)
2.
∫ b
a[f (x) + g(x)] dx =
∫ b
af (x) dx +
∫ b
ag(x) dx.
3.
∫ b
acf (x) dx = c
∫ b
af (x) dx.
4.
∫ b
a[f (x)− g(x)] dx =
∫ b
af (x) dx −
∫ b
ag(x) dx.
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
1.
∫ b
ac dx = c(b − a)
2.
∫ b
a[f (x) + g(x)] dx =
∫ b
af (x) dx +
∫ b
ag(x) dx.
3.
∫ b
acf (x) dx = c
∫ b
af (x) dx.
4.
∫ b
a[f (x)− g(x)] dx =
∫ b
af (x) dx −
∫ b
ag(x) dx.
More Properties of the Integral
Conventions: ∫ a
bf (x) dx = −
∫ b
af (x) dx
∫ a
af (x) dx = 0
This allows us to have
5.
∫ c
af (x) dx =
∫ b
af (x) dx +
∫ c
bf (x) dx for all a, b, and c .
More Properties of the Integral
Conventions: ∫ a
bf (x) dx = −
∫ b
af (x) dx
∫ a
af (x) dx = 0
This allows us to have
5.
∫ c
af (x) dx =
∫ b
af (x) dx +
∫ c
bf (x) dx for all a, b, and c .
More Properties of the Integral
Conventions: ∫ a
bf (x) dx = −
∫ b
af (x) dx
∫ a
af (x) dx = 0
This allows us to have
5.
∫ c
af (x) dx =
∫ b
af (x) dx +
∫ c
bf (x) dx for all a, b, and c .
Example
Suppose f and g are functions with
I
∫ 4
0f (x) dx = 4
I
∫ 5
0f (x) dx = 7
I
∫ 5
0g(x) dx = 3.
Find
(a)
∫ 5
0[2f (x)− g(x)] dx
(b)
∫ 5
4f (x) dx .
SolutionWe have
(a) ∫ 5
0[2f (x)− g(x)] dx = 2
∫ 5
0f (x) dx −
∫ 5
0g(x) dx
= 2 · 7− 3 = 11
(b) ∫ 5
4f (x) dx =
∫ 5
0f (x) dx −
∫ 4
0f (x) dx
= 7− 4 = 3
SolutionWe have
(a) ∫ 5
0[2f (x)− g(x)] dx = 2
∫ 5
0f (x) dx −
∫ 5
0g(x) dx
= 2 · 7− 3 = 11
(b) ∫ 5
4f (x) dx =
∫ 5
0f (x) dx −
∫ 4
0f (x) dx
= 7− 4 = 3
Outline
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
Comparison Properties of the Integral
TheoremLet f and g be integrable functions on [a, b].
6. If f (x) ≥ 0 for all x in [a, b], then∫ b
af (x) dx ≥ 0
7. If f (x) ≥ g(x) for all x in [a, b], then∫ b
af (x) dx ≥
∫ b
ag(x) dx
8. If m ≤ f (x) ≤ M for all x in [a, b], then
m(b − a) ≤∫ b
af (x) dx ≤ M(b − a)
Comparison Properties of the Integral
TheoremLet f and g be integrable functions on [a, b].
6. If f (x) ≥ 0 for all x in [a, b], then∫ b
af (x) dx ≥ 0
7. If f (x) ≥ g(x) for all x in [a, b], then∫ b
af (x) dx ≥
∫ b
ag(x) dx
8. If m ≤ f (x) ≤ M for all x in [a, b], then
m(b − a) ≤∫ b
af (x) dx ≤ M(b − a)
Comparison Properties of the Integral
TheoremLet f and g be integrable functions on [a, b].
6. If f (x) ≥ 0 for all x in [a, b], then∫ b
af (x) dx ≥ 0
7. If f (x) ≥ g(x) for all x in [a, b], then∫ b
af (x) dx ≥
∫ b
ag(x) dx
8. If m ≤ f (x) ≤ M for all x in [a, b], then
m(b − a) ≤∫ b
af (x) dx ≤ M(b − a)
Comparison Properties of the Integral
TheoremLet f and g be integrable functions on [a, b].
6. If f (x) ≥ 0 for all x in [a, b], then∫ b
af (x) dx ≥ 0
7. If f (x) ≥ g(x) for all x in [a, b], then∫ b
af (x) dx ≥
∫ b
ag(x) dx
8. If m ≤ f (x) ≤ M for all x in [a, b], then
m(b − a) ≤∫ b
af (x) dx ≤ M(b − a)
Example
Estimate
∫ 2
1
1
xdx using the comparison properties.
SolutionSince
1
2≤ x ≤ 1
1
for all x in [1, 2], we have
1
2· 1 ≤
∫ 2
1
1
xdx ≤ 1 · 1
Example
Estimate
∫ 2
1
1
xdx using the comparison properties.
SolutionSince
1
2≤ x ≤ 1
1
for all x in [1, 2], we have
1
2· 1 ≤
∫ 2
1
1
xdx ≤ 1 · 1