trapcontinuous - 07.02 Trapezoidal Rule After reading this...

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07.02 Trapezoidal Rule After reading this chapter, you should be able to: 1. Understand the Trapezoidal rule of Integration and how to use it to solve problems. What is Integration? Integration is the process of measuring the area under a function plotted on a graph. Why would we want to do so? Among the most common examples are finding the velocity of a body from acceleration functions, and displacement of a body from velocity data. Throughout the engineering fields, there are (what sometimes seems like) countless applications for integral calculus. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods have been developed to find the integral. Here, we will discuss the trapezoidal rule of approximating integrals of the form ( 29 = b a dx x f I where ) ( x f is called the integrand, = a lower limit of integration = b upper limit of integration 07.02.1
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07.02.2 Chapter 07.02 Figure 1: Integration of a function What is the Trapezoidal Rule? Trapezoidal rule is based on the Newton-Cotes formula that if one approximates the integrand by an n th order polynomial, then the integral of the function is approximated by the integral of that n th order polynomial. Integrating polynomials is simple and is based on the calculus formula. 0 , 1 1 1 + - = + + n n a b dx x n n b a n (1) So if we want to approximate the integral = b a dx x f I ) ( (2) to find the value of the above integral, one assumes ) ( ) ( x f x f n (3) where n n n n n x a x a x a a x f + + + + = - - 1 1 1 0 ... ) ( . (4) where ) ( x f n is an th n order polynomial. Trapezoidal rule assumes 1 = n , that is, the area under the linear polynomial (straight line), b a b a dx x f dx x f ) ( ) ( 1
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Trapezoidal Rule 07.02.3 Derivation of the Trapezoidal Rule Method 1: Derived from Calculus Hence b a b a dx x f dx x f ) ( ) ( 1 + = b a dx x a a ) ( 1 0 - + - = 2 ) ( 2 2 1 0 a b a a b a . (5) But what is a 0 and a 1 ? Now if one chooses, )) ( , ( a f a and )) ( , ( b f b as the two points to approximate ) ( x f by a straight line from a to b , a a a a f a f 1 0 1 ) ( ) ( + = = (6) b a a b f b f 1 0 1 ) ( ) ( + = = (7) Solving the above two equations for a and b , a b a f b f a - - = ) ( ) ( 1 a b a b f b a f a - - = ) ( ) ( 0 (8) Hence from Equation (5), 2 ) ( ) ( ) ( ) ( ) ( ) ( 2 2 a b a b a f b f a b a b a b f b a f dx x f b a - - - + - - - = + - = 2 ) ( ) ( ) ( b f a f a b (9) Method 2: Also derived from Calculus ) ( 1 x f can also be approximated by using the Newton’s divided difference polynomial as ) ( ) ( ) ( ) ( ) ( 1 a x a b a f b f a f x f - - - + = (10)
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07.02.4 Chapter 07.02 Hence b a b a dx x f dx x f ) ( ) ( 1 - - - + = b a dx a x a b a f b f a f ) ( ) ( ) ( ) ( b a ax x a b a f b f x a f -
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trapcontinuous - 07.02 Trapezoidal Rule After reading this...

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