Lecture 16
●
MEEN 357 Homework Solution
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Lecture16HomeworkSolution2009.doc
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1.
Problem 17.3 of the textbook
Solution:
This problem is essentially the same as Problem 17.2 that you worked on the homework
for Lecture 15.
You are asked to evaluate the integral
()
2
0
63
c
o
s
I
xd
x
π
=+
∫
(16.1)
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
c
o
s
f
xx
is elementary and is given by
( ) ()
c
o
s
6 3
s
i
n
x
dx
x
x
C
+=
+
+
∫
(16.2)
Therefore,
2
2
0
0
c
o
s
s
i
n
6
3
s
i
n
3 3
22
12.4248
x
a
x
Ix
d
x
x
x
ππ
=
=
⎛⎞
=
+
=
+
=
+
⎜⎟
⎝⎠
=
∫
(16.3)
Part b)
Single application of trapezoidal rule
The formula for the approximate integral is
( ) ( )
2
b
f
af
b
Ib
a
+
=−
(16.4)
Since
( ) ( )
09
6
2
fa
f
fb
f
=
=
=
=
(16.5)
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MEEN 357 Homework Solution
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it follows from (16.4) that
()
( ) ( )
96 1
5
11.78
22
2
4
b
fa
fb
Ib
a
π
+
⎛⎞
+
=−
=
=
=
⎜⎟
⎝⎠
(16.6)
and true error is
12.4248 11.78
100
100
5.18%
12.4248
ab
t
a
II
I
ε
−
−
==
=
(16.7)
Part c)
Composite trapezoidal rule with
2
N
=
and
4
N
=
.
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 Fall '07
 ANAMALAI
 Composite Simpson, single application, true error

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