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Lecture 15
●
MEEN 357 Homework Solution
1
Lecture15HomeworkSolution2009.doc
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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
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MEEN 357 Homework Solution
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Lecture15HomeworkSolution2009.doc
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()
( ) ( )
0 .9997
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 Fall '07
 ANAMALAI

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