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04/21/23http://
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Trapezoidal Rule of Integration
Civil Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
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Trapezoidal Rule of Integration
http://numericalmethods.eng.usf.edu
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What is IntegrationIntegration:
b
a
dx)x(fI
The process of measuring the area under a function plotted on a graph.
Where:
f(x) is the integrand
a= lower limit of integration
b= upper limit of integration
f(x)
a b
b
a
dx)x(f
y
x
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Basis of Trapezoidal Rule
b
a
dx)x(fI
Trapezoidal Rule is based on the Newton-Cotes Formula that states if one can approximate the integrand as an nth order polynomial…
where )x(f)x(f n
nn
nnn xaxa...xaa)x(f
1110and
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Basis of Trapezoidal Rule
b
an
b
a
)x(f)x(f
Then the integral of that function is approximated by the integral of that nth order polynomial.
Trapezoidal Rule assumes n=1, that is, the area under the linear polynomial,
2)b(f)a(f
)ab(b
a
dx)x(f
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Derivation of the Trapezoidal Rule
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Method Derived From Geometry
The area under the curve is a trapezoid. The integral
trapezoidofAreadxxfb
a
)(
)height)(sidesparallelofSum(2
1
)ab()a(f)b(f 2
1
2)b(f)a(f
)ab(
Figure 2: Geometric Representation
f(x)
a b
b
a
dx)x(f1
y
x
f1(x)
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Example 1
Since decays rapidly as , we will approximate
a) Use single segment Trapezoidal rule to find the value of erfc(0.6560).
b) Find the true error, for part (a).c) Find the absolute relative true error, for part (a).
6560.0
5
2
6560.0 dzeerfc z
The concentration of benzene at a critical location is given by
758.56560.075.1 73.32 erfceerfcc
where dzexerfcx
z
2
So in the above formula dzeerfc z
6560.0
2
6560.0
tE
a
2ze z
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Solution
2
)b(f)a(f)ab(Ia)
2
)( zezf
115 103888.1)5(2 ef
65029.0)6560.0(26560.0 ef
5a
6560.0b
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Solution (cont)
a)
b)The exact value of the above integral cannot be found. We assume the value obtained by adaptive numerical integration using Maple as the exact value for calculating the true error and relative true error.
4124.1
2
65029.0103888.156560.0
11
I
31333.0
6560.06560.0
5
2
dzeerfc z
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Solution (cont)b) ValueeApproximatValueTrueEt
c) The absolute relative true error, , would be t
)4124.1(31333.0
0991.1
%79.350
10031333.0
4124.131333.0
100Value True
Error True
t
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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.
t.t
ln)t(f 892100140000
1400002000
30
19
19
8
30
8
dt)t(fdt)t(fdt)t(f
2
30191930
2
198819
)(f)(f)(
)(f)(f)(
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Multiple Segment Trapezoidal Rule
With
s/m.)(f 271778
s/m.)(f 7548419
s/m.)(f 6790130
2
67.90175.484)1930(
2
75.48427.177)819()(
30
8
dttf
m11266
Hence:
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Multiple Segment Trapezoidal Rule
1126611061 tE
m205
The true error is:
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.
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Multiple Segment Trapezoidal Rule
f(x)
a b
y
x
4
aba
4
2ab
a
4
3ab
a
Figure 4: Multiple (n=4) Segment Trapezoidal Rule
Divide into equal segments as shown in Figure 4. Then the width of each segment is:
n
abh
The integral I is:
b
a
dx)x(fI
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b
h)n(a
h)n(a
h)n(a
ha
ha
ha
a
dx)x(fdx)x(f...dx)x(fdx)x(f1
1
2
2
http://numericalmethods.eng.usf.edu16
Multiple Segment Trapezoidal Rule
The integral I can be broken into h integrals as:
b
a
dx)x(f
Applying Trapezoidal rule on each segment gives:
b
a
dx)x(f
)b(f)iha(f)a(f
n
ab n
i
1
1
22
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Example 2
Since decays rapidly as , we will approximate
a) Use two-segment Trapezoidal rule to find the value of erfc(0.6560).
b) Find the true error, for part (a).c) Find the absolute relative true error, for part (a).
6560.0
5
2
6560.0 dzeerfc z
The concentration of benzene at a critical location is given by 758.56560.075.1 73.32 erfceerfcc
where dzexerfcx
z
2
So in the above formula dzeerfc z
6560.0
2
6560.0
atE
2ze z
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Solutiona) The solution using 2-segment Trapezoidal rule is
)()(2)(2
1
1
bfihafafn
abI
n
i
2n
5a
6560.0b
172.22
56560.0
n
abh
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Solution (cont)Then:
70695.0
65029.000033627.02103888.14
344.4
6560.0828.2254
344.4
6560.02522
56560.0
11
12
1
fff
fihaffIi
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Solution (cont)b) The exact value of the above integral cannot be found. We assume the value obtained by adaptive numerical integration using Maple as the exact value for calculating the true error and relative true error.
so the true error is
39362.0
70695.031333.0
ValueeApproximatValueTrueEt
31333.06560.06560.0
5
2
dzeerfc z
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Solution (cont)
c)The absolute relative true error, , would be t
%63.125
10031333.0
70695.031333.0
100Value True
Error True
t
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Solution (cont)
Table 1 gives the values obtained using
multiple segment Trapezoidal rule for:
Table 1: Multiple Segment Trapezoidal Rule Values
6560.0
5
2
6560.0 dzeerfc z
%t %an Value Et
1 −1.4124 1.0991 350.79 ---
2 −0.70695 0.39362 125.63 99.793
3 −0.48812 0.17479 55.787 44.829
4 −0.40571 0.092379 29.483 20.314
5 −0.37028 0.056957 18.178 9.5662
6 −0.35212 0.038791 12.380 5.1591
7 −0.34151 0.028182 8.9946 3.1063
8 −0.33475 0.021426 6.8383 2.0183
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Example 3
Use Multiple Segment Trapezoidal Rule to find the area under the curve
xe
x)x(f
1
300from to 0x 10x
.
Using two segments, we get 52
010 h
01
03000
0
e
)()(f 03910
1
53005
5.
e
)()(f
1360
1
1030010
10.
e
)()(f
and
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Solution
)b(f)iha(f)a(f
n
abI
n
i
1
12
2
)(f)(f)(f
)( i105020
22
010 12
1
)(f)(f)(f 10520
4
10
136003910204
10.).(
53550.
Then:
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Solution (cont)
So what is the true value of this integral?
592461
30010
0
.dxe
xx
Making the absolute relative true error:
%.
..t 100
59246
5355059246
%.50679
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Solution (cont)
n Approximate Value
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%
Table 2: Values obtained using Multiple Segment Trapezoidal Rule for:
10
01
300dx
e
xx
tE t
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http://numericalmethods.eng.usf.edu27
Error in Multiple Segment Trapezoidal Rule
The true error for a single segment Trapezoidal rule is given by:
ba),("f)ab(
Et
12
3
where is some point in b,a
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
haa),("f
a)ha(E
11
3
1 12
)("fh
1
3
12
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Error in Multiple Segment Trapezoidal Rule
Similarly:
ihah)i(a),("f
)h)i(a()iha(E iii
1
12
1 3
)("fh
i12
3
It then follows that:
bh)n(a),("f
h)n(abE nnn
1
12
1 3
)("fh
n12
3
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http://numericalmethods.eng.usf.edu29
Error in Multiple Segment Trapezoidal Rule
Hence the total error in multiple segment Trapezoidal rule is
n
iit EE
1
n
ii )("f
h
1
3
12 n
)("f
n
)ab(
n
ii
1
2
3
12
The term n
)("fn
ii
1is an approximate average value of the bxa),x("f
Hence:
n
)("f
n
)ab(E
n
ii
t
12
3
12
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Error in Multiple Segment Trapezoidal Rule
Below is the table for the integral
30
8
892100140000
1400002000 dtt.
tln
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.
tE %t %an Value
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 ResourcesFor 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