2. Express the following limits as a definite integral of a suitable function over a finite interval. (a) lim 1ート 1* + 2k +...+ nk nk+1 k>0 (b) lim In
2. Express the following limits as a definite integral of a suitable function over a finite interval. (a) lim 1ート 1* + 2k +...+ nk nk+1 k>0 (b) lim In
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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Please solve all subparts completely
![1. Consider the function f defined by
if r € (0,1]
f(r) =
2- if r e [1,3]
Let P, be the partition of [0,3] into n subintervals of equal length.
(a) Compute L(f, P.) and U(f, P.).
L(f, P.) +U(f, Pn)
(b) Define K(f, P.) =
Show that
2
K(f, P.)-
| S(z)dz <U(f, P.) – L(f, P.).
2. Express the following limits as a definite integral of a suitable function over a finite interval.
1* + 2k +...+n*
(a) lim
,k>0
nk+1
(b) lim In
3. For each of the following choose a suitable partition of the interval of integration and verify the given
inequalities.
(a) e2 -e< Inz dr < 2e2 - e.
/4
(b) /
sec r dr <
-/4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4654d3a-a61f-43c1-b797-b818a0cf5d64%2F859ef51a-baee-4dc4-9015-88b80f1d3dea%2Fqp8q4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Consider the function f defined by
if r € (0,1]
f(r) =
2- if r e [1,3]
Let P, be the partition of [0,3] into n subintervals of equal length.
(a) Compute L(f, P.) and U(f, P.).
L(f, P.) +U(f, Pn)
(b) Define K(f, P.) =
Show that
2
K(f, P.)-
| S(z)dz <U(f, P.) – L(f, P.).
2. Express the following limits as a definite integral of a suitable function over a finite interval.
1* + 2k +...+n*
(a) lim
,k>0
nk+1
(b) lim In
3. For each of the following choose a suitable partition of the interval of integration and verify the given
inequalities.
(a) e2 -e< Inz dr < 2e2 - e.
/4
(b) /
sec r dr <
-/4
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