mws_gen_int_ppt_trapcontinuous(1)

mws_gen_int_ppt_trapcontinuous(1) - Trapezoidal Rule of...

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1/10/2010 http://numericalmethods.eng.usf.edu 1 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
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Trapezoidal Rule of Integration http://numericalmethods.eng.usf.edu
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http://numericalmethods.eng.usf.edu 3 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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http://numericalmethods.eng.usf.edu 4 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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http://numericalmethods.eng.usf.edu 5 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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http://numericalmethods.eng.usf.edu 6 Derivation of the Trapezoidal Rule
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http://numericalmethods.eng.usf.edu 7 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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http://numericalmethods.eng.usf.edu 8 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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http://numericalmethods.eng.usf.edu 9 Solution + 2 ) b ( f ) a ( f ) a b ( I a) 8 = a 30 = b t . t
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mws_gen_int_ppt_trapcontinuous(1) - Trapezoidal Rule of...

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