Transcript Trapezoidal Rule Integration

Trapezoidal Rule of Integration
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
7/21/2015
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1
Trapezoidal Rule of
Integration
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What is Integration
b
Integration:
 f ( x )dx
a
y
f(x)
The process of measuring
the area under a function
plotted on a graph.
b
I   f ( x )dx
a
Where:
f(x) is the integrand
a= lower limit of integration
b= upper limit of integration
3
a
b
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x
Basis of Trapezoidal Rule
Trapezoidal Rule is based on the Newton-Cotes
Formula that states if one can approximate the
integrand as an nth order polynomial…
b
I   f ( x )dx
where
f ( x )  fn( x )
a
and
4
f n ( x )  a0  a1 x  ...  an1 x n1  an x n
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Basis of Trapezoidal Rule
Then the integral of that function is approximated
by the integral of that nth order polynomial.
b
b
a
a
 f ( x )   fn( x )
Trapezoidal Rule assumes n=1, that is, the area
under the linear polynomial,
b

a
5
 f ( a )  f ( b )

(
b

a
)
f ( x )dx


2
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Derivation of the Trapezoidal Rule
6
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Method Derived From Geometry
b
The area under the
curve is a trapezoid.
The integral
 f ( x )dx
1
a
y
f(x)
b
 f ( x)dx  Area of trapezoid
a
f1(x)
1
 ( Sum of parallel sides )( height )
2

1
 f ( b )  f ( a )( b  a )
2
 f ( a )  f ( b )
 ( b  a )

2

a
b
Figure 2: Geometric Representation
7
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x
Example 1
The vertical distance covered by a rocket from t=8 to
t=30 seconds is given by:
140000




x    2000 ln 

9
.
8
t
dt

140000  2100t 

8
30
a) Use single segment Trapezoidal rule to find the distance covered.
b) Find the true error, Et for part (a).
c) Find the absolute relative true error, a for part (a).
8
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Solution
a)
 f ( a )  f ( b )
I  ( b  a )

2

a 8
b  30
140000


f ( t )  2000 ln 
 9.8t

140000  2100t 
140000


f ( 8 )  2000 ln 
 9.8( 8 )

140000  2100( 8 )
 177.27 m / s
140000


f ( 30 )  2000 ln 
 9.8( 30 )  901.67 m / s

140000  2100( 30 )
9
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Solution (cont)
177.27  901.67 
I  ( 30  8 )

2

a)
 11868 m
b) The exact value of the above integral is
140000




x    2000 ln 

9
.
8
t
dt

140000  2100t 

8
30
10
 11061 m
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Solution (cont)
b)
Et  True Value  Approximate Value
 11061  11868
 807 m
c)
The absolute relative true error, t , would be
11061  11868
t 
 100  7.2959%
11061
11
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Multiple Segment Trapezoidal Rule
In Example 1, the true error using single segment trapezoidal rule was
large. We can divide the interval [8,30] into [8,19] and [19,30] intervals
and apply Trapezoidal rule over each segment.
140000


f ( t )  2000ln
  9.8t
 140000  2100t 
30
19
30
8
8
19
 f ( t )dt   f ( t )dt   f ( t )dt
 f ( 8 )  f ( 19 )
 f ( 19 )  f ( 30 )
 ( 19  8 )
 ( 30  19 )


2
2



12
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Multiple Segment Trapezoidal Rule
With
f ( 8 )  177.27 m / s
f ( 30 )  901.67 m / s
f ( 19 )  484.75 m / s
Hence:
30

8
177.27  484.75
 484.75  901.67
f (t )dt  (19  8) 

(
30

19
)



2
2

 11266 m
13
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Multiple Segment Trapezoidal Rule
The true error is:
Et  11061 11266
 205 m
The true error now is reduced from -807 m to -205 m.
Extending this procedure to divide the interval into equal
segments to apply the Trapezoidal rule; the sum of the
results obtained for each segment is the approximate
value of the integral.
14
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Multiple Segment Trapezoidal Rule
y
f(x)
Divide into equal segments
as shown in Figure 4. Then
the width of each segment is:
h
ba
n
The integral I is:
b
I   f ( x )dx
a
a
a
ba
4
a2
ba
4
a3
ba
4
b
Figure 4: Multiple (n=4) Segment Trapezoidal Rule
15
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x
Multiple Segment Trapezoidal Rule
The integral I can be broken into h integrals as:
b
ah
a  2h
a  ( n 1 )h
a
a
ah
a  ( n  2 )h
 f ( x )dx   f ( x )dx   f ( x )dx  ... 
 f ( x )dx 
b
 f ( x )dx
a  ( n 1 )h
Applying Trapezoidal rule on each segment gives:
b

a
16
ba

n 1


f
(
a
)

2
f
(
a

ih
)

f
(
b
)
f ( x )dx




2n 
 i 1

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Example 2
The vertical distance covered by a rocket from to seconds is
given by:
140000




x    2000ln 
 9.8t dt

140000  2100t 

8
30
a) Use two-segment Trapezoidal rule to find the distance covered.
b) Find the true error, Et for part (a).
c) Find the absolute relative true error, a for part (a).
17
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Solution
a) The solution using 2-segment Trapezoidal rule is
ba

n 1

I
f
(
a
)

2
f
(
a

ih
)

f
(
b
)




2n 
 i 1


n2
a 8
ba
30  8
h

n
2
18
b  30
 11
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Solution (cont)
Then:
30  8 

21

I
f
(
8
)

2
f
(
a

ih
)

f
(
30
)




2( 2 ) 
 i 1



22
 f ( 8 )  2 f ( 19 )  f ( 30 )
4

22
177.27  2( 484.75 )  901.67
4
 11266 m
19
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Solution (cont)
b) The exact value of the above integral is
140000




x    2000 ln 

9
.
8
t
dt  11061 m

140000  2100t 

8
30
so the true error is
Et  True Value  Approximate Value
 11061  11266
20
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Solution (cont)
The absolute relative true error, t , would be
t 

T rueError
 100
T rueValue
11061 11266
 100
11061
 1.8534 %
21
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Solution (cont)
Table 1 gives the values
obtained using multiple
segment Trapezoidal rule for
30
140000




x    2000ln

9
.
8
t
dt

140000

2100
t



8
Exact Value=11061 m
n
Value
Et
t %
a %
1
11868
-807
7.296
---
2
11266
-205
1.853
5.343
3
11153
-91.4
0.8265
1.019
4
11113
-51.5
0.4655
0.3594
5
11094
-33.0
0.2981
0.1669
6
11084
-22.9
0.2070
0.09082
7
11078
-16.8
0.1521
0.05482
8
11074
-12.9
0.1165
0.03560
Table 1: Multiple Segment Trapezoidal Rule Values
22
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Example 3
Use Multiple Segment Trapezoidal Rule to find
the area under the curve
300x
f(x)
1 ex
from
Using two segments, we get
f(0)
23
300( 0 )
0
0
1 e
f (5) 
x0
to
x  10
10  0
h
5
2
300( 5 )
 10.039
5
1 e
f ( 10 ) 
and
300( 10 )
 0.136
10
1 e
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Solution
Then:

n 1

ba
I
f
(
a
)

2
f
(
a

ih
)

f
(
b
)




2n 
 i 1



2 1

10  0 

f
(
0
)

2
f
(
0

5
)

f
(
10
)




2( 2 ) 
 i 1



10
 f ( 0 )  2 f ( 5 )  f ( 10 )  10 0  2( 10.039 )  0.136
4
4
 50.535
24
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Solution (cont)
So what is the true value of this integral?
10

300 x
01 e
x
dx  246.59
Making the absolute relative true error:
t 
246.59  50.535
 100%
246.59
 79.506%
25
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Solution (cont)
Table 2: Values obtained using Multiple Segment
10
Trapezoidal Rule for:
300 x

01 e
26
x
dx
n
Approximate
Value
Et
1
0.681
245.91
99.724%
2
50.535
196.05
79.505%
4
170.61
75.978
30.812%
8
227.04
19.546
7.927%
16
241.70
4.887
1.982%
32
245.37
1.222
0.495%
64
246.28
0.305
0.124%
t
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Error in Multiple Segment
Trapezoidal Rule
The true error for a single segment Trapezoidal rule is given by:
( b  a )3
Et 
f " (  ), a    b
12
where 
is some point in a ,b
What is the error, then in the multiple segment Trapezoidal rule? It will
be simply the sum of the errors from each segment, where the error in
each segment is that of the single segment Trapezoidal rule.
The error in each segment is
E1 
27
( a  h )  a3
12
h3

f " ( 1 )
12
f " ( 1 ), a  1  a  h
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Error in Multiple Segment
Trapezoidal Rule
Similarly:
Ei 
( a  ih )  ( a  ( i  1 )h )3
12
f " (  i ), a  ( i  1 )h   i  a  ih
h3

f " ( i )
12
It then follows that:
En 
b  a  ( n  1 )h3
12
f " (  n ), a  ( n  1 )h   n  b
h3

f" ( n )
12
28
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Error in Multiple Segment
Trapezoidal Rule
Hence the total error in multiple segment Trapezoidal rule is
n
n
3
h3 n
(
b

a
)

 f " ( i ) 
12 i 1
12n 2
Et   Ei
i 1
n
The term

i 1
f " ( i )
 f " ( i )
i 1
n
is an approximate average value of the f " ( x ), a  x  b
n
Hence:
n
Et 
29
(b  a )
12n 2
3
 f " ( i )
i 1
n
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Error in Multiple Segment
Trapezoidal Rule
Below is the table for the integral
30
140000




2000
ln

9
.
8
t

dt

140000  2100t 

8
as a function of the number of segments. You can visualize that as the
number of segments are doubled, the true error gets approximately quartered.
30
n
Value
Et
t %
a %
2
11266
-205
1.854
5.343
4
11113
-51.5
0.4655
0.3594
8
11074
-12.9
0.1165
0.03560
16
11065
-3.22
0.02913
0.00401
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Additional Resources
For all resources on this topic such as digital audiovisual
lectures, primers, textbook chapters, multiple-choice
tests, worksheets in MATLAB, MATHEMATICA, MathCad
and MAPLE, blogs, related physical problems, please
visit
http://numericalmethods.eng.usf.edu/topics/trapezoidal
_rule.html
THE END
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