L4_NumericalIntegration - MA2213 Lecture 4 Numerical...

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MA2213 Lecture 4 Numerical Integration
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Introduction Definition is the limit of Riemann sums = b a x d x I ) ( f ) f ( http://www.slu.edu/classes/maymk/Riemann/Riemann.html http://en.wikipedia.org/wiki/Riemann_sum I(f) is called an integral and the process of calculating it is called integration – it has an enormous range of applications http://www.intmath.com/Applications-integration/Applications-integrals-intro.php
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Method of Exhaustion http://en.wikipedia.org/wiki/Method_of_exhaustion was used in ancient times to compute areas and volumes of standard geometric objects Example: area of region between the x-axis, the graph of a function y = f(x), and the vertical lines x = a, and x = b, is given by = b a x d x I ) ( f ) f ( http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/area.html
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Fundamental Theorem of Calculus (Newton and Leibniz) implies that ) ( ) ( ) ( f ) f ( a F b F x d x I b a = = where F is any antiderivative of f, this means that ] , [ ), ( f ) ( b a x x x dx dF = Unfortunately, not all integrands f have ‘closed form’ antiderivatives ) exp( ) ( f 2 x x =
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Left Riemann Sum [f(x 0 ) + f(x 1 ) + . .. + f(x n-1 )] *Delta x
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Right Riemann Sum [f(x 1 ) + f(x 2 ) + . .. + f(x n )] * Delta x http://mathews.ecs.fullerton.edu/a2001/Animations/Quadrature/Midpoint/Midpointaa.html Midpoint Rule Animation
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Midpoint Rule [f(m 1 ) + f(m 2 ) + . .. + f(m n )] * Delta x 2 , , 2 , 2 1 2 1 2 1 0 1 n n n x x m x x m x x m + = + = + = L Animation http://mathews.ecs.fullerton.edu/a2001/Animations/Quadrature/Midpoint/Midpointaa.html
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Trapezoidal Rule The trapezoid approximation associated with a uniform partition a = x0 < x1 < . .. < xn = b is given by .5*[f(x0) + 2f(x1) + . .. + 2f(xn-1) + f(xn)]*Delta x
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Review Questions How can the trapezoidal rule be obtained from the left and right Riemann sums ?
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This note was uploaded on 01/06/2012 for the course MA 2213 taught by Professor Michael during the Fall '07 term at National University of Singapore.

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L4_NumericalIntegration - MA2213 Lecture 4 Numerical...

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