Ch E 310 - Fall 10 - Lecture 19

Ch E 310 - Fall 10 - Lecture 19 - Lecture 19 November 2,...

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Lecture 19 – November 2, 2010 Agenda: Exam 2 returned Thursday (average ~74.7) Complete HW group evaluation (see email) Numerical Integration Newton-Cotes Formulas o Trapezoidal Rule o Simpson’s Rule MATLAB integration techniques
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Numerical Integration Integration is ubiquitous in science and engineering – you already understand its importance Calculating analytical integrals accurately can be challenging – many functions don’t have analytical solutions Many times we can only do so numerically since we can’t always determine analytical expressions Often we are integrating data points rather than functions – akin to curve fitting Depending on the application, certain methods will be more applicable or attractive than others
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Newton-Cotes Formulas The Newton-Cotes formulas are the most common numerical integration schemes They replace a complicated function or discrete data with a polynomial that is easier to evaluate: where f n ( x ) is an n th order polynomial:     dx x f x f I b a n b a   n n n x a x a x a a x f ... 2 2 1 0
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Newton-Cotes Formulas The integrating function can be polynomials of any order: (a) straight lines (b) parabolas The integral can be approximated in one step or in a series of steps to improve accuracy Closed form: curve includes all data between a and b Open form: curve extends beyond data (extrapolation)
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Trapezoidal Rule The trapezoidal rule is an implementation of the Newton- Cotes formula using a first order polynomial (straight line) Note the form: I = (width) × (average height) This form is general for any Newton-Cotes formula               2 ) ( b f a f a b I dx a x a b a f b f a f I dx x f I b a b a n
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Composite Trapezoidal Rule The trapezoidal rule applied to the entire interval is usually not accurate enough, however if we divide the interval into smaller segments, it’s much more useful If we have n + 1 data points, there are n segments We evaluate each segment and sum them up       n n i i x f x f x f h I 1 1 0 2 2 n a b h Segment intervals: Note: this applies to equally- spaced data points
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Simpson’s Rules: 1/3 Rule We can also increase the order of the polynomial to improve the integral estimate If we have three points, this can be fit to a parabola (remember polynomial interpolation?): I
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This note was uploaded on 09/21/2011 for the course CH E 310 taught by Professor Staff during the Spring '08 term at Iowa State.

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Ch E 310 - Fall 10 - Lecture 19 - Lecture 19 November 2,...

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