6) a) Let the function f been given by S1, f(x) = {0, when 0 < x < 1 when x = 1 Show that for each e > 0 there is a partition P, of [0,1] such that for the lower Riemann sum L(f, P.) applies that L(f, P.) > 1 – €.
6) a) Let the function f been given by S1, f(x) = {0, when 0 < x < 1 when x = 1 Show that for each e > 0 there is a partition P, of [0,1] such that for the lower Riemann sum L(f, P.) applies that L(f, P.) > 1 – €.
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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6a
![6)
a) Let the function f been given by
(1,
when 0 < x < 1
f(x) =
(o,
when x = 1
Show that for each e > 0 there is a partition P, of [0,1] such that for the lower
Riemann sum L(f,P.) applies that L(f, P.) > 1 - €.
b) Use (a) to show that f is integrable and that f(x)dx = 1
c) Let g be a decreasing continuous function on [0,1], assuming the values g(0) =
1 and g(1) = 0. Show that there is a partition P of [0,1] such that the following
inequalities apply to the lower and the upper Riemann sum.
0 < L(g, P)< U(g,P) < 1
d) Use c) to show the inequalities
0 <
g(x)dx < 1
(You can use that a continuous function on [0,1] is integrable)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78f3f25b-2eb2-4186-b342-f9f487e8326d%2Ffd0a1cf2-756f-41f3-8071-19e3b6f58729%2F208rutf_processed.png&w=3840&q=75)
Transcribed Image Text:6)
a) Let the function f been given by
(1,
when 0 < x < 1
f(x) =
(o,
when x = 1
Show that for each e > 0 there is a partition P, of [0,1] such that for the lower
Riemann sum L(f,P.) applies that L(f, P.) > 1 - €.
b) Use (a) to show that f is integrable and that f(x)dx = 1
c) Let g be a decreasing continuous function on [0,1], assuming the values g(0) =
1 and g(1) = 0. Show that there is a partition P of [0,1] such that the following
inequalities apply to the lower and the upper Riemann sum.
0 < L(g, P)< U(g,P) < 1
d) Use c) to show the inequalities
0 <
g(x)dx < 1
(You can use that a continuous function on [0,1] is integrable)
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