Lecture 15
●
MEEN 357 Homework Solution
1
Lecture15HomeworkSolution2009.doc
RMB
1.
Problem 17.2 of the textbook.
This problem should be turned in, as usual, at the start of class.
Solution:
You are to evaluate the integral
( )
4
2
0
1
x
I
ed
x
−
=−
∫
(15.1)
a)
Analytically
b)
Single application of trapezoidal rule.
c)
Composite trapezoidal rule with
2
N
=
and
4
N
=
.
d)
Single application of Simpson’s 1/3 rule.
e)
Composite Simpson’s 1/3 rule with
4
N
=
.
f)
Simpson’s 3/8 rule.
The book does not specify a single application of the 3/8
rule.
For simplicity, we shall work the single application case.
You are asked to determine the percent relative error for parts b) through f) based upon
the answer to part a).
Part a)
Exact
The integral of
()
2
1
x
f
xe
−
is elementary and is given by
22
1
1
2
xx
x
x
e
C
−−
−
=+
+
∫
(15.2)
Therefore,
4
4
228
8
0
0
11
1
7
1
14
2
2
2
3.5002
x
a
x
I
e
e
e
=
−−−
−
=
⎛⎞
=
−
= +
−=+
⎜⎟
⎝⎠
=
∫
(15.3)
Part b)
Single application of trapezoidal rule
The formula for the approximate integral is
( ) ( )
2
b
f
af
b
Ib
a
+
(15.4)
Since
( ) ( )
() ()
00
4
.9997
fa
f
fb
f
==
(15.5)
it follows from (15.4) that
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentLecture 15
●
MEEN 357 Homework Solution
2
Lecture15HomeworkSolution2009.doc
RMB
()
( ) ( )
0 .9997
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 ANAMALAI
 single application, true error

Click to edit the document details