• Riemann's definition: f is integrable on [a, b] if and only if for some I e R, for any ɛ > 0, there is & > 0 such that for any partition P : a = xo < ¤1 < • ·< xn =bwith |P| = max |x; – xi-1| < 8, any choices of Ef(&)(#1 – #i-1) – 1|< e. i=1 • Darboux's definition: f is integrable on [a, b] if and only if f is bounded and the upper integral equals to lower integral, i.e., U(f) = infU(f, P) = L(f) = sup L(f, P). P Show that these two definitions are equivalent using the Darboux's Theorem: U(f) = lim U(f, P), |P|→0 L(f) = lim L(f, P). |P]¬0 dint: from Darboux's definition to Riemann's, it is basically an application of the squeeze theorem; from Riemann's lefinition to Darboux's, use the definition of inf and sup to approximate any lower/upper sums by Riemann sums. Be careful that you cannot assume f is continuous, so the suprema and infima may not be achieved!
• Riemann's definition: f is integrable on [a, b] if and only if for some I e R, for any ɛ > 0, there is & > 0 such that for any partition P : a = xo < ¤1 < • ·< xn =bwith |P| = max |x; – xi-1| < 8, any choices of Ef(&)(#1 – #i-1) – 1|< e. i=1 • Darboux's definition: f is integrable on [a, b] if and only if f is bounded and the upper integral equals to lower integral, i.e., U(f) = infU(f, P) = L(f) = sup L(f, P). P Show that these two definitions are equivalent using the Darboux's Theorem: U(f) = lim U(f, P), |P|→0 L(f) = lim L(f, P). |P]¬0 dint: from Darboux's definition to Riemann's, it is basically an application of the squeeze theorem; from Riemann's lefinition to Darboux's, use the definition of inf and sup to approximate any lower/upper sums by Riemann sums. Be careful that you cannot assume f is continuous, so the suprema and infima may not be achieved!
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
How do you solve this? Typed answers are preferred but not necessary, thank you! Will upvote answer!
![Recall that we have two definitions for integrability.
• Riemann's definition: f is integrable on [a, b] if and only if for some I E R, for any ɛ > 0, there is & > 0 such
that for any partition P : a = xo < x1 < • .< xn = b with P= max r; – xi–1 < 8, any choices of
i
&i E [Ti, ti-1],
- Xi-1
E.
i=1
• Darboux's definition: f is integrable on [a, b| if and only if f is bounded and the upper integral equals to lower
integral, i.e.,
U(f) = infU(f, P) = L(f)
sup L(f, P).
P
Show that these two definitions are equivalent using the Darboux's Theorem:
U(f) = lim U(f, P),
|P|¬0
L(f) = lim L(f, P).
|P|→0
Hint: from Darboux's definition to Riemann's, it is basically an application of the squeeze theorem; from Riemann's
definition to Darboux's, use the definition of inf and sup to approximate any lower/upper sums by Riemann sums.
Be careful that you cannot assume f is continuous, so the suprema and infima may not be achieved!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8c2ad84-3927-490a-bf66-2d9eb7f56b39%2Ff740b1c4-e34e-4623-a45f-d1d5859739a2%2Fjrvxam2_processed.png&w=3840&q=75)
Transcribed Image Text:Recall that we have two definitions for integrability.
• Riemann's definition: f is integrable on [a, b] if and only if for some I E R, for any ɛ > 0, there is & > 0 such
that for any partition P : a = xo < x1 < • .< xn = b with P= max r; – xi–1 < 8, any choices of
i
&i E [Ti, ti-1],
- Xi-1
E.
i=1
• Darboux's definition: f is integrable on [a, b| if and only if f is bounded and the upper integral equals to lower
integral, i.e.,
U(f) = infU(f, P) = L(f)
sup L(f, P).
P
Show that these two definitions are equivalent using the Darboux's Theorem:
U(f) = lim U(f, P),
|P|¬0
L(f) = lim L(f, P).
|P|→0
Hint: from Darboux's definition to Riemann's, it is basically an application of the squeeze theorem; from Riemann's
definition to Darboux's, use the definition of inf and sup to approximate any lower/upper sums by Riemann sums.
Be careful that you cannot assume f is continuous, so the suprema and infima may not be achieved!
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
