ECE3040 Chapter 7.pdf - Chapter 7 The lecture notes are selected from the public Powerpoint lecture documents provided by Authors Autar Kaw Sri Harsha

ECE3040 Chapter 7.pdf - Chapter 7 The lecture notes are...

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Chapter 7 The lecture notes are selected from the public Powerpoint lecture documents provided by Authors: Autar Kaw, Sri Harsha Garapati
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Trapezoidal Rule of Integration
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What is Integration Integration: b a dx ) x ( f I 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 ( f I Trapezoidal Rule is based on the Newton-Cotes Formula that states if one can approximate the integrand as an n th order polynomial… where ) x ( f ) x ( f n n n n n n x a x a ... x a a ) x ( f 1 1 1 0 and
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Basis of Trapezoidal Rule b a n b a ) x ( f ) x ( f Then the integral of that function is approximated by the integral of that n th order polynomial. Trapezoidal Rule assumes n=1, that is, the area under the linear polynomial, 2 ) b ( f ) a ( f ) a b ( 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 trapezoid of Area dx x f b a ) ( ) height )( sides parallel of Sum ( 2 1 ) a b ( ) a ( f ) b ( f 2 1 2 ) b ( f ) a ( f ) a b ( Figure 2: Geometric Representation f(x) a b b a dx ) x ( f 1 y x f 1 (x)
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Example 1 The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: 30 8 8 9 2100 140000 140000 2000 dt t . t ln x a) Use single segment Trapezoidal rule to find the distance covered. b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a). t E a
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Solution 2 ) b ( f ) a ( f ) a b ( I a) 8 a 30 b t . t ln ) t ( f 8 9 2100 140000 140000 2000 ) ( . ) ( ln ) ( f 8 8 9 8 2100 140000 140000 2000 8 ) ( . ) ( ln ) ( f 30 8 9 30 2100 140000 140000 2000 30 s / m . 27 177 s / m . 67 901
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Solution (cont) 2 67 901 27 177 8 30 . . ) ( I m 11868 a) b) The exact value of the above integral is 30 8 8 9 2100 140000 140000 2000 dt t . t ln x m 11061
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Solution (cont) b) Value e Approximat Value True E t 11868 11061 m 807 c) The absolute relative true error , , would be t 100 11061 11868 11061 t % . 2959 7
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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 8 9 2100 140000 140000 2000 30 19 19 8 30 8 dt ) t ( f dt ) t ( f dt ) t ( f 2 30 19 19 30 2 19 8 8 19 ) ( f ) ( f ) ( ) ( f ) ( f ) (
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Multiple Segment Trapezoidal Rule With s / m . ) ( f 27 177 8 s / m . ) ( f 75 484 19 s / m . ) ( f 67 901 30 2 67 . 901 75 . 484 ) 19 30 ( 2 75 . 484 27 . 177 ) 8 19 ( ) ( 30 8 dt
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