CP06_integration - Integration S = f ( x )dx a b Part 1...

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1 1 Integration = b a dx x f S ) ( Ö Exact integration Ö Simple numerical methods Ö Advanced numerical methods 2 Part 1 Exact integration 3 Three possible ways for exact integration Ö Standard techniques of integration substitution rule, integration by parts, using identities, … Ö Tables of integrals Ö Computer algebra systems 4 Tables of integrals Table of Integrals, Series and Products by Gradshteyn I. S. and Ryzhik I. M. Academic Press, 1994 (many editions since 195x) (most referenced in physics) Integral and Series , vol.1-3, by Prudnikov A P, Brychkov Yu A and Marichev A I Gordon and Breach, New York, 1986 (most sophisticated) Tables of Integrals and Other Mathematical Data by Herbert B. Dwight (very simple integrals) and many more 5 Computer algebra systems Ö Maple Ö Mathematica Ö MathCad Ö Scientific Workplace Ö Derive 6 Part 2 Basic ideas
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2 7 Quite often we need numerical integrations Ö if you can not get an analytic answer using Tables of integrals Computer algebra systems … and various calculus books Ö if you have a discrete set of data points, i.e. as a result of measurements or calculations ) ( i i x f 8 Integrating approximating functions Numerical integration can be based on fitting approximating functions (polynomials) to discrete data and integrating approximating functions = b a n b a dx x P dx x f I ) ( ) ( 9 Integrating approximating functions Case 1: The function to be integrated is known only at a finite set of discrete points Parameters under control – the degree of approximating polynomial 10 Integrating approximating functions Case 2: The function to be integrated is known. Parameters under control Ö The total number of discrete points Ö The degree of the approximating polynomial to represent the discrete data. Ö The locations of the points at which the known function is discretized 11 Part 3a Direct fit polynomials 12 Direct fit polynomials The procedure can be used for both unequally and equally spaced data It is based on fitting the data by a direct fit polynomial and integrating that polynomial. b a b a n b a n x a x a x a dx x P dx x f I x a x a a x P x f + + + = = + + + = K K 3 2 ) ( ) ( ) ( ) ( 3 2 2 1 0 2 2 1 0 then
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3 13 Part 3b Quadrature methods on equal subintervals 14 Riemann Integral If is a continuous function defined for and we divide the interval into n subintervals of equal width then the definite integral is Bernhard Riemann, 1826-1866, German mathematician = Δ = n i i n b a x x f dx x f 1 * ) ( lim ) ( ) ( x f b x a n a b x / ) ( = Δ The Riemann integral can be interpreted as a net area under the curve from a to b ) ( x f y = 15 Three simples integration methods 1. Left endpoint Riemann sum n a b x x x f L dx x f n i i n n b a / ) ( , ) ( lim ) ( 1 1 = Δ Δ = = = 16 Three simples integration methods (cont.) 2. Right endpoint Riemann sum n a b x x x f R dx x f n i i n n b a / ) ( , ) ( lim ) ( 1 = Δ Δ = = = 17 Three simples integration methods (cont.) 3. Midpoint Riemann sum n a b x x x x f R dx x f n i i i n n b a / ) ( , ) 2 ( lim ) ( 1 1 = Δ Δ + = =
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CP06_integration - Integration S = f ( x )dx a b Part 1...

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