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Lecture Note of Unequal Segments Trapezoidal Method

الكلية كلية الهندسة     القسم  الهندسة البيئية     المرحلة 3
أستاذ المادة وليد علي حسن       13/03/2017 07:16:53
University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)








Unequal 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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2.1 – Unequal Segments Trapezoidal Method
The integration of a function with unequal segments can be approximately calculated using the trapezoid method as follows:

??x?_1+??x?_2+?+??x?_n=(b-a)
I=?_a^b??f(x)dx=??x?_1 (y_0+y_1)/2+??x?_2 (y_1+y_2)/2+?+??x?_n (y_(n-1)+y_n)/2?
??x?_1+??x?_2+?+??x?_n=(b-a)
?=|(True Value-Approximate Value)/(True Value)|×100%

where:
?: the error value.
x_0=a y_0=f(x_0)
x_1=x_0+??x?_1 y_1=f(x_1)
x_n=x_(n-1)+??x?_n=b y_n=f(x_n)

Ex1: Use the trapezoidal method to evaluate the value of the following integration.
?_0^4???xe?^2x dx?
??x?_1=2, ??x?_2=1, ??x?_3=0.5,and ??x?_4=0.5
Compare your solution with the exact solution (I=5216.92).

Solution:
a=0 , b=4
Four unequal segments
y_i=f(x_i )=?xe?^2x
i x y
0 x_0=a=0 ?(0)×e?^(2×(0))=0
1 ?0+?x?_1=2 ?(2)×e?^(2×(2))=109.2
2 2+??x?_2=3 ?(3)×e?^(2×(3))=1210.3
3 ?3+?x?_3=3.5 ?(3.5)×e?^(2×(3.5))=3838.2
4 ?3.5+?x?_3=4=b ?(4)×e?^(2×(4))=11923.8

I=?_0^4???xe?^2x dx?=??x?_1 (y_0+y_1)/2+??x?_2 (y_1+y_2)/2+?+??x?_n (y_(n-1)+y_n)/2=2×(0+109.2)/2+1×(109.2+1210.3)/2+0.5×(1210.3+3838.2)/2+0.5×(3838.2+11923.8)/2=5971.55

Compare with the true value:
?=|(True Value-Approximate Value)/(True Value)|×100%=|(5216.92-5971.55)/5216.92|×100%=14.5%

Ex2: Use the trapezoidal method to evaluate the value of the following integration.
?_0^10??300x/(1+e^x ) dx?
Use two unequal segments : ??x?_1=4, ??x?_2=6
Compare your solution with the exact solution (I=246.59).

Solution:

a=0 , b=10
Two un equal segments
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+?x?_1=0+4=4 (300×(4))/(1+e^((4)) )=21.58
2 ?4+?x?_2=4+6=10=b (300×(10))/(1+e^((10)) )=0.14

I=?_0^10??300x/(1+e^x ) dx?=??x?_1 (y_0+y_1)/2+??x?_2 (y_1+y_2)/2+?+??x?_n (y_(n-1)+y_n)/2=4×(0+21.58)/2+6×(21.58+0.14)/2=138.32

Compare with the true value:
?=|(True Value-Approximate Value)/(True Value)|×100%=|(246.59-138.32)/246.59|×100%=43.5%

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Homework 12
Use the unequal trapezoidal method to estimate the value of the following integration.
?_0^2??x^2 dx?
The number of segments (n) is equal to 3 as: (??x?_1=0.5, ??x?_2=0.5,and ??x?_3=1)
Compare your solution with the exact solution.
Use the unequal trapezoidal method to estimate the value of the following integration.
?_0^4???xe?^2x dx?
The number of segments (n) is equal to 3 as:
(??x?_1=1, ??x?_2=1,and ??x?_3=2)
Compare your solution with the exact solution (I=5216.92).


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