Advanced Calculus (I)
WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
5.1 Riemann Integral
Definition
Let a, b ∈R with a < b.
(i)
A partition of the interval [a, b] is a set of points P = {x0,x1, . . . ,xn} such that
a = x0<x1< · · · <xn =b.
5.1 Riemann Integral
Definition
Let a, b ∈R with a < b.
(i)
A partition of the interval [a, b] is a set of points P = {x0,x1, . . . ,xn} such that
a = x0<x1< · · · <xn =b.
5.1 Riemann Integral
Definition
Let a, b ∈R with a < b.
(i)
A partition of the interval [a, b] is a set of points P = {x0,x1, . . . ,xn} such that
a = x0<x1< · · · <xn =b.
5.1 Riemann Integral
Definition
Let a, b ∈R with a < b.
(i)
A partition of the interval [a, b] is a set of points P = {x0,x1, . . . ,xn} such that
a = x0<x1< · · · <xn =b.
(ii)
The norm of a partition P = {x0,x1, . . . ,xn} is the number
||P|| = max
1≤j≤n|xj − xj−1|.
Definition (ii)
The norm of a partition P = {x0,x1, . . . ,xn} is the number
||P|| = max
1≤j≤n|xj − xj−1|.
(iii)
A refinement of a partition P = {x0,x1, . . . ,xn} is a partition Q of [a,b] that satisfies Q ⊃ P. In this case we say that Q is finer than P.
Definition (iii)
A refinement of a partition P = {x0,x1, . . . ,xn} is a partition Q of [a,b] that satisfies Q ⊃ P. In this case we say that Q is finer than P.
(iii)
A refinement of a partition P = {x0,x1, . . . ,xn} is a partition Q of [a,b] that satisfies Q ⊃ P. In this case we say that Q is finer than P.
Definition
Let a, b ∈R with a < b, let P = {x0,x1, . . . ,xn} be a partition of the interval [a,b], and suppose that f : [a, b] →R is bounded.
(i)
The upper Riemann sum of f over P is the number
U(f , P) :=
n
X
j=1
Mj(f )(xj − xj−1),
where
Mj(f ) := sup
x ∈[xj−1,xj]
f (x )
Let a, b ∈R with a < b, let P = {x0,x1, . . . ,xn} be a partition of the interval [a,b], and suppose that f : [a, b] →R is bounded.
(i)
The upper Riemann sum of f over P is the number
U(f , P) :=
n
X
j=1
Mj(f )(xj − xj−1),
where
Mj(f ) := sup
x ∈[xj−1,xj]
f (x )
Definition
Let a, b ∈R with a < b, let P = {x0,x1, . . . ,xn} be a partition of the interval [a,b], and suppose that f : [a, b] →R is bounded.
(i)
The upper Riemann sum of f over P is the number
U(f , P) :=
n
X
j=1
Mj(f )(xj − xj−1),
where
Mj(f ) := sup
x ∈[xj−1,xj]
f (x )
Let a, b ∈R with a < b, let P = {x0,x1, . . . ,xn} be a partition of the interval [a,b], and suppose that f : [a, b] →R is bounded.
(i)
The upper Riemann sum of f over P is the number
U(f , P) :=
n
X
j=1
Mj(f )(xj − xj−1),
where
Mj(f ) := sup
x ∈[xj−1,xj]
f (x )
Definition (ii)
The lower Riemann sum of f over P is the number
L(f , P) :=
n
X
j=1
mj(f )(xj − xj−1)
where
mj(f ) := inf
x ∈[xj−1,xj]f (x ).
[Note: We assumed that f is bounded so that the numbers Mj(f ) and mj(f ) would exist and be finite.]
(ii)
The lower Riemann sum of f over P is the number
L(f , P) :=
n
X
j=1
mj(f )(xj − xj−1)
where
mj(f ) := inf
x ∈[xj−1,xj]f (x ).
[Note: We assumed that f is bounded so that the numbers Mj(f ) and mj(f ) would exist and be finite.]
Definition (ii)
The lower Riemann sum of f over P is the number
L(f , P) :=
n
X
j=1
mj(f )(xj − xj−1)
where
mj(f ) := inf
x ∈[xj−1,xj]f (x ).
[Note: We assumed that f is bounded so that the numbers Mj(f ) and mj(f ) would exist and be finite.]
(ii)
The lower Riemann sum of f over P is the number
L(f , P) :=
n
X
j=1
mj(f )(xj − xj−1)
where
mj(f ) := inf
x ∈[xj−1,xj]f (x ).
[Note: We assumed that f is bounded so that the numbers Mj(f ) and mj(f ) would exist and be finite.]
Lemma:
L(f , P) ≤ U(f , P)for all partitions P and all bounded functions f.
Lemma:
L(f , P) ≤ U(f , P)for all partitions P and all bounded functions f.
Lemma:
If Pand Q are any partitions of [a,b], then L(f , P) ≤ U(f , Q).
Lemma:
If Pand Q are any partitions of [a,b], then L(f , P) ≤ U(f , Q).
Definition
Let a, b ∈R with a < b. A function f : [a, b] → R is said to be (Riemann) integrable on [a,b], if and only if f is
bounded on [a,b] and for every > 0 there is a partition P of [a,b] such that U(f , P) − L(f , P) < .
Let a, b ∈R with a < b. A function f : [a, b] → R is said to be (Riemann) integrable on [a,b], if and only if f is
bounded on [a,b] and for every > 0 there is a partition P of [a,b] such that U(f , P) − L(f , P) < .
Theorem
Suppose that a, b ∈R with a < b. If f is continuous on the interval [a,b], then f is integrable on [a,b].
Suppose that a, b ∈R with a < b. If f is continuous on the interval [a,b], then f is integrable on [a,b].
Proof:
Let > 0. Since f is uniformly continuous on [a,b],choose δ >0 such that
(1) |x − y | < δ implies |f (x) − f (y )| < b − a. Let P = {x0,x1, . . . ,xn} be any partition of [a,b] that satisfies ||P|| < δ. Fix an index j and notice, by the Extreme Value Theorem, that there are points xm and xM
in [xj−1,xj]such that
f (xm) =mj(f ) and f (xM) =Mj(f ).
Proof:
Let > 0.Since f is uniformly continuous on [a,b], choose δ >0 such that
(1) |x − y | < δ implies |f (x) − f (y )| < b − a. Let P = {x0,x1, . . . ,xn} be any partition of [a,b] that satisfies ||P|| < δ. Fix an index j and notice, by the Extreme Value Theorem, that there are points xm and xM
in [xj−1,xj]such that
f (xm) =mj(f ) and f (xM) =Mj(f ).
Proof:
Let > 0. Since f is uniformly continuous on [a,b],choose δ >0 such that
(1) |x − y | < δ implies |f (x) − f (y )| < b − a. Let P = {x0,x1, . . . ,xn} be any partition of [a,b] that satisfies ||P|| < δ. Fix an index j and notice, by the Extreme Value Theorem, that there are points xm and xM
in [xj−1,xj]such that
f (xm) =mj(f ) and f (xM) =Mj(f ).
Proof:
Let > 0. Since f is uniformly continuous on [a,b], choose δ >0 such that
(1) |x − y | < δ implies |f (x) − f (y )| < b − a. Let P = {x0,x1, . . . ,xn} be any partition of [a,b] that satisfies ||P|| < δ. Fix an index j and notice,by the Extreme Value Theorem, that there are points xm and xM
in [xj−1,xj]such that
f (xm) =mj(f ) and f (xM) =Mj(f ).
Proof:
Let > 0. Since f is uniformly continuous on [a,b], choose δ >0 such that
(1) |x − y | < δ implies |f (x) − f (y )| < b − a. Let P = {x0,x1, . . . ,xn} be any partition of [a,b] that satisfies ||P|| < δ. Fix an index j and notice, by the Extreme Value Theorem,that there are points xm and xM
in [xj−1,xj]such that
f (xm) =mj(f ) and f (xM) =Mj(f ).
Proof:
Let > 0. Since f is uniformly continuous on [a,b], choose δ >0 such that
(1) |x − y | < δ implies |f (x) − f (y )| < b − a. Let P = {x0,x1, . . . ,xn} be any partition of [a,b] that satisfies ||P|| < δ. Fix an index j and notice,by the Extreme Value Theorem, that there are points xm and xM
in [xj−1,xj]such that
f (xm) =mj(f ) and f (xM) =Mj(f ).
Proof:
Let > 0. Since f is uniformly continuous on [a,b], choose δ >0 such that
(1) |x − y | < δ implies |f (x) − f (y )| < b − a. Let P = {x0,x1, . . . ,xn} be any partition of [a,b] that satisfies ||P|| < δ. Fix an index j and notice, by the Extreme Value Theorem,that there are points xm and xM
in [xj−1,xj]such that
f (xm) =mj(f ) and f (xM) =Mj(f ).
Proof:
Let > 0. Since f is uniformly continuous on [a,b], choose δ >0 such that
(1) |x − y | < δ implies |f (x) − f (y )| < b − a. Let P = {x0,x1, . . . ,xn} be any partition of [a,b] that satisfies ||P|| < δ. Fix an index j and notice, by the Extreme Value Theorem, that there are points xm and xM
in [xj−1,xj]such that
f (xm) =mj(f ) and f (xM) =Mj(f ).
Since ||P|| < δ, we also have |xM − xm| < δ. Hence by (1), Mj(f ) − mj(f ) <
(b − a). In particular,
U(f , P)−L(f , P) =
n
X
j=1
(Mj(f )−mj(f ))(xj−xj−1) < b − a
n
X
j=1
(xj−xj−1) = .
(The last step comes from telescoping.) 2
Since ||P|| < δ,we also have |xM m
Mj(f ) − mj(f ) <
(b − a). In particular,
U(f , P)−L(f , P) =
n
X
j=1
(Mj(f )−mj(f ))(xj−xj−1) < b − a
n
X
j=1
(xj−xj−1) = .
(The last step comes from telescoping.) 2
Since ||P|| < δ, we also have |xM − xm| < δ. Hence by (1), Mj(f ) − mj(f ) <
(b − a). In particular,
U(f , P)−L(f , P) =
n
X
j=1
(Mj(f )−mj(f ))(xj−xj−1) < b − a
n
X
j=1
(xj−xj−1) = .
(The last step comes from telescoping.) 2
Since ||P|| < δ, we also have |xM m
Mj(f ) − mj(f ) <
(b − a). In particular,
U(f , P)−L(f , P) =
n
X
j=1
(Mj(f )−mj(f ))(xj−xj−1) < b − a
n
X
j=1
(xj−xj−1) = .
(The last step comes from telescoping.) 2
Since ||P|| < δ, we also have |xM − xm| < δ. Hence by (1), Mj(f ) − mj(f ) <
(b − a). In particular,
U(f , P)−L(f , P) =
n
X
j=1
(Mj(f )−mj(f ))(xj−xj−1) < b − a
n
X
j=1
(xj−xj−1) = .
(The last step comes from telescoping.) 2
Since ||P|| < δ, we also have |xM m
Mj(f ) − mj(f ) <
(b − a). In particular,
U(f , P)−L(f , P) =
n
X
j=1
(Mj(f )−mj(f ))(xj−xj−1) < b − a
n
X
j=1
(xj−xj−1) = .
(The last step comes from telescoping.) 2
Since ||P|| < δ, we also have |xM − xm| < δ. Hence by (1), Mj(f ) − mj(f ) <
(b − a). In particular,
U(f , P)−L(f , P) =
n
X
j=1
(Mj(f )−mj(f ))(xj−xj−1) < b − a
n
X
j=1
(xj−xj−1) = .
(The last step comes from telescoping.) 2
Example:
The function
f (x ) =
0 0 ≤ x < 1 2
1 1
2 ≤ x ≤ 1 is integrable on [0,1].
Example:
The function
f (x ) =
0 0 ≤ x < 1 2
1 1
2 ≤ x ≤ 1 is integrable on [0,1].
Let a, b ∈R with a < b, and f : [a, b] → R be bounded.
(i)
The upper integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := inf{U(f , P) : P is a partition of [a, b]}
(ii)
The lower integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := sup{L(f , P) : P is a partition of [a, b]}
Definition
Let a, b ∈R with a < b, and f : [a, b] → R be bounded.
(i)
The upper integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := inf{U(f , P) : P is a partition of [a, b]}
(ii)
The lower integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := sup{L(f , P) : P is a partition of [a, b]}
Let a, b ∈R with a < b, and f : [a, b] → R be bounded.
(i)
The upper integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := inf{U(f , P) : P is a partition of [a, b]}
(ii)
The lower integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := sup{L(f , P) : P is a partition of [a, b]}
Definition
Let a, b ∈R with a < b, and f : [a, b] → R be bounded.
(i)
The upper integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := inf{U(f , P) : P is a partition of [a, b]}
(ii)
The lower integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := sup{L(f , P) : P is a partition of [a, b]}
Let a, b ∈R with a < b, and f : [a, b] → R be bounded.
(i)
The upper integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := inf{U(f , P) : P is a partition of [a, b]}
(ii)
The lower integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := sup{L(f , P) : P is a partition of [a, b]}
Definition
Let a, b ∈R with a < b, and f : [a, b] → R be bounded.
(i)
The upper integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := inf{U(f , P) : P is a partition of [a, b]}
(ii)
The lower integral of f on [a,b] is the number
(U) Z b
a
f (x )dx := sup{L(f , P) : P is a partition of [a, b]}
(iii)
If the upper and lower integrals of f on [a,b] are equal, we define the integral of f on [a,b] to be the common value
Z b a
f (x )dx := (U) Z b
a
f (x )dx = (L) Z b
a
f (x )dx .
Definition (iii)
If the upper and lower integrals of f on [a,b] are equal, we define the integral of f on [a,b] to be the common value
Z b a
f (x )dx := (U) Z b
a
f (x )dx = (L) Z b
a
f (x )dx .
(iii)
If the upper and lower integrals of f on [a,b] are equal, we define the integral of f on [a,b] to be the common value
Z b a
f (x )dx := (U) Z b
a
f (x )dx = (L) Z b
a
f (x )dx .
Remark:
If f : [a, b] →R is bounded, then its upper and lower integrals exists and are infinite, and satisfy
(L) Z b
a
f (x )dx ≤ (U) Z b
a
f (x )dx .
Remark:
If f : [a, b] →R is bounded, then its upper and lower integrals exists and are infinite, and satisfy
(L) Z b
a
f (x )dx ≤ (U) Z b
a
f (x )dx .
Theorem
Let a, b ∈R with a < b, and f : [a, b] → R be bounded.
Then f is integrable on [a,b] if and only if
(L) Z b
a
f (x )dx = (U) Z b
a
f (x )dx .
Let a, b ∈R with a < b, and f : [a, b] → R be bounded.
Then f is integrable on [a,b] if and only if
(L) Z b
a
f (x )dx = (U) Z b
a
f (x )dx .
