Lecture Note of Equal Segments Trapezoidal Method

University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)








Equal Segments Trapezoidal Method Undergraduate Level, 3th Stage



Mr. Waleed Ali Tameemi
Engineer/ College of Engineering/ Babylon University
M.Sc. Civil Engineering/ the University of Kansas/ USA



2016-2017
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1.0 – Introduction
Integration can be defined as the area under the curve of a function plotted on a graph. The following sections will discuss numerical methods of integration.















I=?_a^b?f(x)dx
where:
I: the Integration value,
f(x): the integrand function,
a: lower limit of integration,
b: upper limit of integration.?
2.0 – Trapezoidal Method
2.1 – Equal Segments Trapezoidal Method
The integration of a function can be approximately calculated using the trapezoid method by estimating the area under the curve of the given function and as follows:

?_a^b??f(x)dx=?x (y_0+y_1)/2+?x (y_1+y_2)/2+?+?x (y_(n-1)+y_n)/2?
?_a^b??f(x)dx=?x/2[y_0+y_n+?_(i=1)^(n-1)???2y?_i]??
?x=(b-a)/n
?=|(True Value-Approximate Value)/(True Value)|×100%
where:
n: the number of segments (the higher the n, the more accurate solution),
?: the error value.
x_0=a y_0=f(x_0)
x_1=x_0+?x y_1=f(x_1)
x_n=x_(n-1)+?x=b y_n=f(x_n)

Ex1: Calculate the value of the given integration.
?_8^30??{2000 ln??[140000/(140000-2100x)?]-9.8x}dx?
The number of segments (n) is equal to 2,4, and 8.
Compare your solution with the exact solution (I=11061).

Solution:

a=8 , b=30

First, two segments (n=2)
?x=(b-a)/n=(30-8)/2=11
y_i=f(x_i )=2000 ln??[140000/(140000-2100x)?]-9.8x_i
i x y
0 x_0=a=8 2000×ln??[140000/(140000-2100×(8))?]-9.8×(8)=177.27

1 8+11=19 2000×ln??[140000/(140000-2100×(19))?]-9.8×(19)=484.8
2 19+11=30=b 2000×ln??[140000/(140000-2100×(30))?]-9.8×(30)=901.7

I=?x/2[y_0+y_n+?_(i=1)^(n-1)??2y?_i ]=11/2[177.27+901.67+2×484.75]=11266.4

Compare with the true value:
?=|(True Value-Approximate Value)/(True Value)|×100%=|(11061-11266.4)/11061|×100%=1.86%

Second, four segments (n=4)
?x=(b-a)/n=(30-8)/4=5.5
y_i=f(x_i )=2000 ln??[140000/(140000-2100x)?]-9.8x_i
i x y
0 x_0=a=8 2000×ln??[140000/(140000-2100×(8))?]-9.8×(8)=177.3

1 8+5.5=13.5 2000×ln??[140000/(140000-2100×(13.5))?]-9.8×(13.5)=320.2
2 13.5+5.5=19 2000×ln??[140000/(140000-2100×(19))?]-9.8×(19)=484.7
3 19+5.5=24.5 2000×ln??[140000/(140000-2100×(24.5))?]-9.8×(24.5)=676.1
4 24.4+5.5=30=b 2000×ln??[140000/(140000-2100×(30))?]-9.8×(30)=901.7

I=?x/2[y_0+y_n+?_(i=1)^(n-1)??2y?_i ]=5.5/2[177.3+901.7+2×(320.2+484.75+676.1)]=11113

Compare with the true value:
?=|(True Value-Approximate Value)/(True Value)|×100%=|(11061-11113)/11061|×100%=0.47%

Third, eight segments (n=8)
?x=(b-a)/n=(30-8)/8=2.75
y_i=f(x_i )=2000 ln??[140000/(140000-2100x)?]-9.8x_i
i x y
0 x_0=a=8 2000×ln??[140000/(140000-2100×(8))?]-9.8×(8)=177.3

1 8+2.75=10.75 2000×ln??[140000/(140000-2100×(10.75))?]-9.8×(10.75)=246.3

2 10.75+2.75=13.5 2000×ln??[140000/(140000-2100×(13.5))?]-9.8×(13.5)=320.2
3 13.5+2.75=16.25 2000×ln??[140000/(140000-2100×(16.25))?]-9.8×(16.25)=399.5
4 16.25+2.75=19 2000×ln??[140000/(140000-2100×(19))?]-9.8×(19)=484.7
5 19+2.75=21.75 2000×ln??[140000/(140000-2100×(21.75))?]-9.8×(21.75)=576.6
6 21.75+2.75=24.5 2000×ln??[140000/(140000-2100×(24.5))?]-9.8×(24.5)=676.1
7 24.5+2.75=27.25 2000×ln??[140000/(140000-2100×(27.25))?]-9.8×(27.25)=784
8 27.25+2.75=30=b 2000×ln??[140000/(140000-2100×(30))?]-9.8×(30)=901.7

I=?x/2[y_0+y_n+?_(i=1)^(n-1)??2y?_i ]=2.75/2[177.3+901.7+2×(246.3+320.2+399.5+484.75+576.6+676.1+784)]=11074

Compare with the true value:
?=|(True Value-Approximate Value)/(True Value)|×100%=|(11061-11074)/11061|×100%=0.118%
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Ex2: Calculate the value of the given integration.
?_0^10??300x/(1+e^x ) dx?
The number of segments (n) is equal to 2,4, and 8.
Compare your solution with the exact solution (I=246.59).

Solution:

a=0 , b=10

First, two segments (n=2)
?x=(b-a)/n=(10-0)/2=5
y_i=f(x_i )=?_0^10??300x/(1+e^x ) dx?
i x y
0 x_0=a=0 (300×(0))/(1+e^((0)) )=0
1 0+5=5 (300×(5))/(1+e^((5)) )=10.04
2 5+5=10=b (300×(10))/(1+e^((10)) )=0.14

I=?x/2[y_0+y_n+?_(i=1)^(n-1)??2y?_i ]=5/2[0+0.14+2×10.04]=50.55

Compare with the true value:
?=|(True Value-Approximate Value)/(True Value)|×100%=|(246.59-50.55)/246.59|×100%=79.5%

Second, four segments (n=4)
?x=(b-a)/n=(10-0)/4=2.5
y_i=f(x_i )=?_0^10??300x/(1+e^x ) dx?
i x y
0 x_0=a=0 (300×(0))/(1+e^((0)) )=0
1 0+2.5=2.5 (300×(2.5))/(1+e^((2.5)) )=56.89
2 2.5+2.5=5 (300×(5))/(1+e^((5)) )=10.04
3 5+2.5=7.5 (300×(7.5))/(1+e^((7.5)) )=1.24
4 7.5+2.5=10=b (300×(10))/(1+e^((10)) )=0.14

I=?x/2[y_0+y_n+?_(i=1)^(n-1)??2y?_i ]=2.5/2[0+0.14+2×(56.89+10.04+1.24)]=170.6

Compare with the true value:
?=|(True Value-Approximate Value)/(True Value)|×100%=|(246.59-170.6)/246.59|×100%=30.82%

Third, eight segments (n=8)
?x=(b-a)/n=(10-0)/8=1.25
y_i=f(x_i )=?_0^10??300x/(1+e^x ) dx?
i x y
0 x_0=a=0 (300×(0))/(1+e^((0)) )=0
1 0+1.25=1.25 (300×(1.25))/(1+e^((1.25)) )=83.51
2 1.25+1.25=2.5 (300×(2.5))/(1+e^((2.5)) )=56.89
3 2.5+1.25=3.75 (300×(3.75))/(1+e^((3.75)) )=25.85
4 3.75+1.25=5 (300×(5))/(1+e^((5)) )=10.04
5 5+1.25=6.25 (300×(6.25))/(1+e^((6.25)) )=3.61
6 6.25+1.25=7.5 (300×(7.5))/(1+e^((7.5)) )=1.24
7 7.5+1.25=8.75 (300×(8.75))/(1+e^((8.75)) )=0.42
8 8.75+1.25=10=b (300×(10))/(1+e^((10)) )=0.14

I=?x/2[y_0+y_n+?_(i=1)^(n-1)??2y?_i ]=1.25/2[0+0.14+2×(83.51+56.89+25.85+10.04+3.61+1.24+0.42)]=227.04

Compare with the true value:
?=|(True Value-Approximate Value)/(True Value)|×100%=|(246.59-227.04)/246.59|×100%=7.93%

|(11061-11266.4)/11061|×100%=1.86%

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Ex3: Calculate the value of the given integration.
?_1^5???(1+x^2 ) dx?
The number of segments (n) is equal to 8.

Solution:

a=1 , b=5

Eight segments (n=8)
?x=(b-a)/n=(5-1)/8=0.5
y_i=f(x_i )=?(1+x^2 )
i x y
0 x_0=a=1 ?(1+?(1)?^2 )=1.41
1 1+0.5=1.5 ?(1+?(1.5)?^2 )=1.80
2 1.5+0.5=2 ?(1+?(2)?^2 )=2.24
3 2+0.5=2.5 ?(1+?(2.5)?^2 )=2.69
4 2.5+0.5=3 ?(1+?(3)?^2 )=3.26
5 3+0.5=3.5 ?(1+?(3.5)?^2 )=3.64
6 3.5+0.5=4 ?(1+?(4)?^2 )=4.12
7 4+0.5=4.5 ?(1+?(4.5)?^2 )=4.61
8 4.5+0.5=5=b ?(1+?(5)?^2 )=5.10

I=?x/2[y_0+y_n+?_(i=1)^(n-1)??2y?_i ]=0.5/2[1.41+5.10+2×(1.80+2.24+2.69+3.26+3.64+4.12+4.61)]=12.81

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Homework 11
Use the trapezoidal method to estimate the value of the following integration.
?_0^2??x^2 dx?
The number of segments (n) is equal to 10.
Compare your solution with the exact solution.
Use the trapezoidal method to estimate the value of the following integration.
?_0^4???xe?^2x dx?
The number of segments (n) is equal to 2 and 8.
Compare your solution with the exact solution (I=5216.92).

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