Trapezoidal Rule of Integration

Trapezoidal Rule of Integration -...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Trapezoidal Rule of  Integration      http://numericalmethods.eng.usf.edu
Background image of page 2
                                            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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
                                            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
Background image of page 4
                                            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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
                                            http://numericalmethods.eng.usf.edu 6 Derivation of the Trapezoidal Rule
Background image of page 6
                                            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)
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
                                            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).
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 32

Trapezoidal Rule of Integration -...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online