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Unformatted text preview: Part Six Numerical Differentiation and Integration Fig PT6.10 Chapter 21 NewtonCotes Integration Formulas Bungee Jumper Given the velocity of the bungee jumper, determine the vertical distance the jumper has fallen after time t Analytic integration (closed form solution) Numerical integration needed for more complex, nonlinear functions t m gc c m t z dt t m gc c gm dt t v t z t m gc c gm t v d d t d d t d d cosh ln ) ( tanh ) ( ) ( tanh ) ( Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral Use of strips to approximate an integral Integration Weighted sum of functional values at specified points NewtonCotes closed or open formulae evenly spaced points Approximate the function by Lagrange interpolation polynomial Integration of a simple interpolation polynomial Guassian Quadratures Basic Numerical Integration Weighted sum of function values ) ( ) ( ) ( ) ( ) ( n n 1 1 i n i i b a x f c x f c x f c x f c dx x f x x 1 x n x n1 x f ( x ) 2 4 6 8 10 12 3 5 7 9 11 13 15 Numerical Integration Idea is to do integral in small parts, like the way you first learned integration  a summation Numerical methods just try to make it faster and more accurate NewtonCotes formulas based on idea dx x f dx x f I b a n b a ) ( ) ( Approximate f ( x ) by a polynomial n n 1 n 1 n 1 n x a x a x a a x f ) ( Numerical integration f n ( x ) can be linear f n ( x ) can be quadratic f n ( x ) can also be cubic or other higherorder polynomials Polynomial can be piecewise over the data Numerical Integration NewtonCotes Closed Formulae  Use both end points Trapezoidal Rule : Linear Simpsons 1/3Rule : Quadratic Simpsons 3/8Rule : Cubic Booles Rule : Fourthorder Higherorder methods NewtonCotes Open Formulae  Use only interior points midpoint rule Higherorder methods Closed and Open Formulae (a) End points are known (b) Extrapolation Trapezoidal Rule Straightline approximation ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 i 1 i i b a x f x f 2 h x f c x f c x f c dx x f x x 1 x f ( x ) L(x) Trapezoidal Rule Lagrange interpolation ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ; , , , ) ( ) ( ) ( b f a f 2 h 2 h b f 2 h a f d h b f d 1 h a f d L h dx x L dx x f b f a f 1 L 1 b x a x a b h h dx d a b a x x b x a let x f x x x x x f x x x x x L 1 2 1 2 1 1 1 b a b a 1 1 1 1 1...
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 Spring '08
 Rottman

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