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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 http://www.slu.edu/classes/maymk/Riemann/Riemann.html = b a x d x I ) ( f ) f ( I(f) is called an integral and the process of calculating it is called integration – it has an enormous range of applications http://en.wikipedia.org/wiki/Riemann_sum http://www.intmath.com/Applications-integration/Applications-integrals-intro.php
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Method of Exhaustion was used in ancient times to compute areas and volumes of standard geometric objects = b a x d x I ) ( f ) f ( http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/area.html 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 http://en.wikipedia.org/wiki/Method_of_exhaustion
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Fundamental Theorem of Calculus (Newton and Leibniz) implies that where F is any antiderivative of f, this means that Unfortunately, not all integrands f have ‘closed form’ antiderivatives ] , [ ), ( f ) ( b a x x x dx dF = ) exp( ) ( f 2 x x - = ) ( ) ( ) ( f ) f ( a F b F x d x I b a - = =
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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 http://mathews.ecs.fullerton.edu/a2001/Animations/Quadrature/Midpoint/Midpointaa.html Animation 2 , , 2 , 2 1 2 1 2 1 0 1 n n n x x m x x m x x m + = + = + = -
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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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