Suppose that f [a, b] → R is strictly increasing, i.e., that if x₁ < x₂ then f(x₁) < f(x₂). Explain : why f is integrable.
Suppose that f [a, b] → R is strictly increasing, i.e., that if x₁ < x₂ then f(x₁) < f(x₂). Explain : why f is integrable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Suppose that f: [a, b] → R is strictly increasing, i.e., that if x₁ < x2 then ƒ(x₁) < f(x2). Explain
why f is integrable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2F0d852377-d610-4911-8289-b741d2882d7b%2Fv84luw_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that f: [a, b] → R is strictly increasing, i.e., that if x₁ < x2 then ƒ(x₁) < f(x2). Explain
why f is integrable.
![Integrability.
:
DEFINITION 7.7. Let f [a,b] → R be a bounded function and let P be the collection of all
partitions of [a, b]. The upper integral of f is defined to be
U(f) = inf{U(f, P) : P ≤ P}.
The lower integral of f is defined to be
L(f) = sup{L(f, P) : P ≤ P}.
LEMMA 7.8. Let f : [a, b] → R be a bounded function, say, m≤ f(x) ≤ M for all x € [a, b]. Then
we have
m(b − a) ≤ L(f) ≤ U(f) ≤ M(b − a).
-
DEFINITION 7.9. A bounded
function f [a, b] → R is integrable if L(f)
=
U(f). When this
rb
happens, we denote
·Sof
f and
afs
f(x) dx to be this common value. That is,
So f(x) dx = L(f) = U(ƒ).
PROPOSITION 7.10. Assume that a bounded function f : [a, b] → R is integrable. If f(x) > 0 for
all x = [a, b], then
S.
f(x)dx ≥ 0.
A Criterion for Integrability.
THEOREM 7.11. Let ƒ : [a,b] → R be bounded. Then f is integrable if and only if for all € > 0
there exists a partition P of [a, b] such that
U(f, Pe) — L(f, Pc) < €.
-
COROLLARY 7.12. ƒ : [a, b] → R is integrable if and only if there exists a sequence P₁ of partitions
of [a, b] such that
lim [U(f, Pn) - L(ƒ, Pn)] = 0.
n→∞](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2F0d852377-d610-4911-8289-b741d2882d7b%2Fahuf3s_processed.png&w=3840&q=75)
Transcribed Image Text:Integrability.
:
DEFINITION 7.7. Let f [a,b] → R be a bounded function and let P be the collection of all
partitions of [a, b]. The upper integral of f is defined to be
U(f) = inf{U(f, P) : P ≤ P}.
The lower integral of f is defined to be
L(f) = sup{L(f, P) : P ≤ P}.
LEMMA 7.8. Let f : [a, b] → R be a bounded function, say, m≤ f(x) ≤ M for all x € [a, b]. Then
we have
m(b − a) ≤ L(f) ≤ U(f) ≤ M(b − a).
-
DEFINITION 7.9. A bounded
function f [a, b] → R is integrable if L(f)
=
U(f). When this
rb
happens, we denote
·Sof
f and
afs
f(x) dx to be this common value. That is,
So f(x) dx = L(f) = U(ƒ).
PROPOSITION 7.10. Assume that a bounded function f : [a, b] → R is integrable. If f(x) > 0 for
all x = [a, b], then
S.
f(x)dx ≥ 0.
A Criterion for Integrability.
THEOREM 7.11. Let ƒ : [a,b] → R be bounded. Then f is integrable if and only if for all € > 0
there exists a partition P of [a, b] such that
U(f, Pe) — L(f, Pc) < €.
-
COROLLARY 7.12. ƒ : [a, b] → R is integrable if and only if there exists a sequence P₁ of partitions
of [a, b] such that
lim [U(f, Pn) - L(ƒ, Pn)] = 0.
n→∞
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