Trapezoidal Rule of Integration

# Trapezoidal Rule of Integration - Major:...

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05/01/11 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
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
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
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 = ( 29 ) 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).
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## This note was uploaded on 05/01/2011 for the course CHBE 2120 taught by Professor Gallivan during the Spring '07 term at Georgia Tech.

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Trapezoidal Rule of Integration - Major:...

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