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Chapter 07.04
Romberg Rule of Integration
After reading this chapter, you should be able to:
1.
derive the Romberg rule of integration, and
2.
use the Romberg rule of integration to solve problems.
What is integration?
Integration is the process of measuring the area under a function plotted on a graph.
Why
would we want to integrate a function?
Among the most common examples are finding the
velocity of a body from an acceleration function, and displacement of a body from a velocity
function.
Throughout many engineering fields, there are (what sometimes seems like)
countless applications for integral calculus.
You can read about some of these applications in
Chapters 07.00A07.00G.
Sometimes, the evaluation of expressions involving these integrals can become daunting, if
not indeterminate.
For this reason, a wide variety of numerical methods has been developed
to simplify the integral.
Here, we will discuss the Romberg rule of approximating integrals of the form
()
∫
=
b
a
dx
x
f
I
(
1
)
where
)
(
x
f
is called the integrand
=
a
lower limit of integration
=
b
upper limit of integration
07.04.1
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Chapter 07.04
Figure 1
Integration of a function.
Error in MultipleSegment Trapezoidal Rule
The true error obtained when using the multiple segment trapezoidal rule with
segments to
approximate an integral
n
()
∫
b
a
dx
x
f
is given by
()
()
n
f
n
a
b
E
n
i
i
t
∑
=
′
′
−
−
=
1
2
3
12
ξ
(
2
)
where for each
i
,
i
is a point somewhere in the domain
( )
[ ]
ih
a
h
i
a
+
−
+
,
1
, and
the term
()
n
f
n
i
i
∑
=
′
′
1
can be viewed as an approximate average value of
( )
x
f
′
′
in
.
This
leads us to say that the true error
in Equation (2) is approximately proportional to
[
b
a
,
]
t
E
2
1
n
E
t
α
≈
(
3
)
for the estimate of
using the
segment trapezoidal rule.
()
∫
b
a
dx
x
f
n
Table 1 shows the results obtained for
dt
t
t
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
⎥
⎦
⎤
⎢
⎣
⎡
−
30
8
8
.
9
2100
140000
140000
ln
2000
using the multiplesegment trapezoidal rule.
Romberg rule of Integration
07.04.3
Table 1
Values obtained using multiple segment trapezoidal rule for
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
⎥
⎦
⎤
⎢
⎣
⎡
−
=
30
8
8
.
9
2100
140000
140000
ln
2000
dt
t
t
x
.
n
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This note was uploaded on 04/30/2010 for the course MAE 107 taught by Professor Rottman during the Fall '08 term at UCSD.
 Fall '08
 Rottman

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