mws_gen_int_txt_simpson13

mws_gen_int_txt_simpson13 - Chapter 07.03 Simpsons 1/3 Rule...

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07.03.1 Chapter 07.03 Simpson’s 1/3 Rule of Integration After reading this chapter, you should be able to 1. derive the formula for Simpson’s 1/3 rule of integration, 2. use Simpson’s 1/3 rule it to solve integrals, 3. develop the formula for multiple-segment Simpson’s 1/3 rule of integration, 4. use multiple-segment Simpson’s 1/3 rule of integration to solve integrals, and 5. derive the true error formula for multiple-segment Simpson’s 1/3 rule. What is integration? Integration is the process of measuring the area under a function plotted on a graph. Why would we want to integrate a function? Among the most common examples are finding the velocity of a body from an acceleration function, and displacement of a body from a velocity function. Throughout many engineering fields, there are (what sometimes seems like) countless applications for integral calculus. You can read about some of these applications in Chapters 07.00A-07.00G. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods has been developed to simplify the integral. Here, we will discuss Simpson’s 1/3 rule of integral approximation, which improves upon the accuracy of the trapezoidal rule. Here, we will discuss the Simpson’s 1/3 rule of approximating integrals of the form  b a dx x f I where ) ( x f is called the integrand, a lower limit of integration b upper limit of integration Simpson’s 1/3 Rule The trapezoidal rule was based on approximating the integrand by a first order polynomial, and then integrating the polynomial over interval of integration. Simpson’s 1/3 rule is an
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07.03.2 Chapter 07.03 extension of Trapezoidal rule where the integrand is approximated by a second order polynomial. Figure 1 Integration of a function Method 1: Hence b a b a dx x f dx x f I ) ( ) ( 2 where ) ( 2 x f is a second order polynomial given by 2 2 1 0 2 ) ( x a x a a x f . Choose )), ( , ( a f a , 2 , 2 b a f b a and )) ( , ( b f b as the three points of the function to evaluate , 0 a 1 a and 2 a . 2 2 1 0 2 ) ( ) ( a a a a a a f a f 2 2 1 0 2 2 2 2 2 b a a b a a a b a f b a f 2 2 1 0 2 ) ( ) ( b a b a a b f b f Solving the above three equations for unknowns, , 0 a 1 a and 2 a give 2 2 2 2 0 2 ) ( ) ( 2 4 ) ( ) ( b ab a a f b a abf b a abf b abf b f a a 2 2 1 2 ) ( 2 4 ) ( 3 ) ( 3 2 4 ) ( b ab a b bf b a bf a bf b af b a af a af a
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Simpson’s 1/3 Rule of Integration 07.03.3 2 2 2 2 ) ( 2 2 ) ( 2 b ab a b f b a f a f a Then
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida.

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mws_gen_int_txt_simpson13 - Chapter 07.03 Simpsons 1/3 Rule...

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