Integrals
Luca Zaffonte
Contents
1 Reimann integral 1
1.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Necessary and sufficient condition for integration according to Riemann . . . . . . . . . . . . . . . . 3
2 Properties of the Riemann integral 4
2.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Comparison property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Absolute comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Integrability unlike a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 Additivity of the integral with respect to the integration domain . . . . . . . . . . . . . . . . . . . . 7
2.5.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.6 Integral additivity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.7 Integral over an oriented interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.8 Peano-Jordan measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.8.1 Integral as a sub-graph area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.9 Integrability of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.10 Integrability of monotone functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Integral function 12
3.1 Mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Fundamental theorem of integral calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 First formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.2 Second formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.3 Exemples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Methods of integration 16
4.1 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Improper integrals 18
5.1 Improper integrals of positive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Criterion of comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Asymptotic comparison criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.4 Integral criterion for the convergence of a series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.5 Absolute convergence criterion for the improper integral . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Fresnel integrals 28
1 Reimann integral
Let I ⊆ R be closed and bounded, I = [a, b], a < b and a, b ∈ R. A subdivision of [a, b] is a set of the type
{x0 , x1 , . . . , xn } ⊆ [a, b] such that a = x0 < xk < xn = b.
1
,Definition: A function f : [a, b] → R is said to scale (or piecewise constant) if there is a subdivision of [a, b] and of
the constants c1 , c2 , . . . , cn ∈ R such that
f (x) = cj ∀xj−1 < x < xj
The set {x0 , . . . , xn } is called a subdivision of [a, b] associated with the f scale function.
Observation: Give two ladder functions f, g : [a, b] → R there exists a subdivision of [a, b] associated with both f
and g.
If {x1 , . . . xn } is a subdivision associated with f and {y1 , . . . , ym } is a subdivision associated with g then the union
of the two is a subdivision for both g and f .
Definition: Let f : [a, b] → R be a ladder function. Let {x1 , . . . xn } be a subdivision associated with f and
c1 , . . . cn ∈ R such that f (x) = cj ∀x ∈ (xj−1 , xj ). The value of f is said to be integral
n
X
cj (xj − xj−1 )
j=1
and it is indicated with
Z Z
f f (x)dx
[a,b] [a,b]
where x is a dumb variable.
Observation: The integral of a ladder function does not depend on the choice of subdivision.
Lemma: Properties of ladder functions’ integral
Let f, g : [a, b] → R scale and let c ∈ R. Then:
f + g is a scale function and
Z Z Z
f +g = f+ g
[a,b] [a,b] [a,b]
c · f is a e scale function
Z Z
c·f =c f
[a,b] [a,b]
If f ≤ g we have that
Z Z
f≤ g
[a,b] [a,b]
|f | is a scale function and
Z Z
f ≤ |f |
[a,b] [a,b]
Z Z
f ≤ |f |
[a,b] [a,b]
Z Z
f ≤ |f |
[a,b] [a,b]
Proof : Fixed {x0 , . . . , xn } a subdivision associated with both g and f , then
f (x) = cj ∀x ∈ (xj−1 , xj ) e g(x) = dj ∀x ∈ (xj−1 , xj )
then
(f + g)(x) = f (x) + g(x) = cj + dj ∀x ∈ (xj−1 , xj )
So the f + g function is a scale function. Moreover
2
, Z n
X
f +g = (cj + dj ) (xj − xj−1 )
[a,b] j=1
Xn n
X
= cj (xj − xj−1 ) + dj (xj − xj−1 )
j=1 j=1
Z Z
= f+ g
[a,b] [a,b]
1.1 Exercise
I suppose that f ≤ g, then we have cj ≤ dj ∀j. Then
Z n
X n
X
f= cj (xj − xj−1 ) ≤ dj (xj − xj−1 )
[a,b] j=1 j=1
∀x ∈ x j−1),xj ) we have |f (x)| = |cj | then |f (x)| is scaled. Moreover
Z n
X n
X Z
f = cj (xj , xj−1 ) ≤ |cj (xj , xj−1 )| = f
[a,b] j=1 j=1 [a,b]
Let f : [a, b] → R bounded (i.e. the image set is bounded). Let them be
(Z )
I− (f ) = S tale che S : [a, b] → R a scala S ≤ f
[a,b]
(Z )
I+ (f ) = S tale che S : [a, b] → R a scala S ≥ f
[a,b]
The assumption that f is bounded guarantees that both are nonempty sets. Furthermore, we have that ∀α ∈ I− (f )
and ∀β ∈ I+ (f ) we have that α ≤ β and I− (f ) is bounded above and I+ (f ) is bounded below. Then sup I− (f )
and inf I+ (f ) are finite and
sup I− (f ) ≤ inf I+ (f )
In particular I call sup I− (f ) lower integral of f on [a, b] and I call inf I+ (f ) higher integral of f on [a, b].
Definition: Let f : [a, b] → R limited. The function f is said to be integrable according to Riemann if
sup I− (f ) = inf I+ (f )
in this case the common value of sup I− (f ) = inf I+ (f ) is said to be integral of f in [a, b].
1.2 Necessary and sufficient condition for integration according to Riemann
Proposition: Let f : [a, b] → R limited. The f function is Riemann integrable if and only if ∀ε > 0 ∃S2 , S2 :
[a, b] → R such that
Z Z
S2 − S1 ≤ ε
[a,b] [a,b]
and
S1 ≤ f ≤ S2
Proof : We begin by proving that if a function is integrable according to Riemann then the two previous statements
hold.
Let f : [a, b] → R integrable according to Riemann, then we know that
3
Luca Zaffonte
Contents
1 Reimann integral 1
1.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Necessary and sufficient condition for integration according to Riemann . . . . . . . . . . . . . . . . 3
2 Properties of the Riemann integral 4
2.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Comparison property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Absolute comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Integrability unlike a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 Additivity of the integral with respect to the integration domain . . . . . . . . . . . . . . . . . . . . 7
2.5.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.6 Integral additivity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.7 Integral over an oriented interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.8 Peano-Jordan measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.8.1 Integral as a sub-graph area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.9 Integrability of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.10 Integrability of monotone functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Integral function 12
3.1 Mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Fundamental theorem of integral calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 First formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.2 Second formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.3 Exemples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Methods of integration 16
4.1 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Improper integrals 18
5.1 Improper integrals of positive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Criterion of comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Asymptotic comparison criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.4 Integral criterion for the convergence of a series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.5 Absolute convergence criterion for the improper integral . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Fresnel integrals 28
1 Reimann integral
Let I ⊆ R be closed and bounded, I = [a, b], a < b and a, b ∈ R. A subdivision of [a, b] is a set of the type
{x0 , x1 , . . . , xn } ⊆ [a, b] such that a = x0 < xk < xn = b.
1
,Definition: A function f : [a, b] → R is said to scale (or piecewise constant) if there is a subdivision of [a, b] and of
the constants c1 , c2 , . . . , cn ∈ R such that
f (x) = cj ∀xj−1 < x < xj
The set {x0 , . . . , xn } is called a subdivision of [a, b] associated with the f scale function.
Observation: Give two ladder functions f, g : [a, b] → R there exists a subdivision of [a, b] associated with both f
and g.
If {x1 , . . . xn } is a subdivision associated with f and {y1 , . . . , ym } is a subdivision associated with g then the union
of the two is a subdivision for both g and f .
Definition: Let f : [a, b] → R be a ladder function. Let {x1 , . . . xn } be a subdivision associated with f and
c1 , . . . cn ∈ R such that f (x) = cj ∀x ∈ (xj−1 , xj ). The value of f is said to be integral
n
X
cj (xj − xj−1 )
j=1
and it is indicated with
Z Z
f f (x)dx
[a,b] [a,b]
where x is a dumb variable.
Observation: The integral of a ladder function does not depend on the choice of subdivision.
Lemma: Properties of ladder functions’ integral
Let f, g : [a, b] → R scale and let c ∈ R. Then:
f + g is a scale function and
Z Z Z
f +g = f+ g
[a,b] [a,b] [a,b]
c · f is a e scale function
Z Z
c·f =c f
[a,b] [a,b]
If f ≤ g we have that
Z Z
f≤ g
[a,b] [a,b]
|f | is a scale function and
Z Z
f ≤ |f |
[a,b] [a,b]
Z Z
f ≤ |f |
[a,b] [a,b]
Z Z
f ≤ |f |
[a,b] [a,b]
Proof : Fixed {x0 , . . . , xn } a subdivision associated with both g and f , then
f (x) = cj ∀x ∈ (xj−1 , xj ) e g(x) = dj ∀x ∈ (xj−1 , xj )
then
(f + g)(x) = f (x) + g(x) = cj + dj ∀x ∈ (xj−1 , xj )
So the f + g function is a scale function. Moreover
2
, Z n
X
f +g = (cj + dj ) (xj − xj−1 )
[a,b] j=1
Xn n
X
= cj (xj − xj−1 ) + dj (xj − xj−1 )
j=1 j=1
Z Z
= f+ g
[a,b] [a,b]
1.1 Exercise
I suppose that f ≤ g, then we have cj ≤ dj ∀j. Then
Z n
X n
X
f= cj (xj − xj−1 ) ≤ dj (xj − xj−1 )
[a,b] j=1 j=1
∀x ∈ x j−1),xj ) we have |f (x)| = |cj | then |f (x)| is scaled. Moreover
Z n
X n
X Z
f = cj (xj , xj−1 ) ≤ |cj (xj , xj−1 )| = f
[a,b] j=1 j=1 [a,b]
Let f : [a, b] → R bounded (i.e. the image set is bounded). Let them be
(Z )
I− (f ) = S tale che S : [a, b] → R a scala S ≤ f
[a,b]
(Z )
I+ (f ) = S tale che S : [a, b] → R a scala S ≥ f
[a,b]
The assumption that f is bounded guarantees that both are nonempty sets. Furthermore, we have that ∀α ∈ I− (f )
and ∀β ∈ I+ (f ) we have that α ≤ β and I− (f ) is bounded above and I+ (f ) is bounded below. Then sup I− (f )
and inf I+ (f ) are finite and
sup I− (f ) ≤ inf I+ (f )
In particular I call sup I− (f ) lower integral of f on [a, b] and I call inf I+ (f ) higher integral of f on [a, b].
Definition: Let f : [a, b] → R limited. The function f is said to be integrable according to Riemann if
sup I− (f ) = inf I+ (f )
in this case the common value of sup I− (f ) = inf I+ (f ) is said to be integral of f in [a, b].
1.2 Necessary and sufficient condition for integration according to Riemann
Proposition: Let f : [a, b] → R limited. The f function is Riemann integrable if and only if ∀ε > 0 ∃S2 , S2 :
[a, b] → R such that
Z Z
S2 − S1 ≤ ε
[a,b] [a,b]
and
S1 ≤ f ≤ S2
Proof : We begin by proving that if a function is integrable according to Riemann then the two previous statements
hold.
Let f : [a, b] → R integrable according to Riemann, then we know that
3
