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NewtonCotes - Closed Newton-Cotes Integration James...

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Closed Newton-Cotes Integration James Keesling This document will discuss Newton-Cotes Integration. Other methods of numerical integration will be discussed in other posts. The other methods will include the Trapezoidal Rule, Romberg Integration, and Gaussian Integration. 1 Principle of Newton-Cotes Integration We only cover the Newton-Cotes closed formulas. The interval of integration [ a, b ] is partitioned by the points. a, a + b - a n , a + 2 · b - a n , . . . , b We estimate the integral of f ( x ) on this interval by using the Lagrange interpolating polynomial through the following points. ( a, f ( a )) , a + b - a n , f a + b - a n , . . . , ( b, f ( b )) The formula for the integral of this Lagrange polynomial simplifies to a linear combi- nation of the values of f ( x ) at the points x i = a + i · b - a n i = 0 , 1 , 2 , . . . , n . In the next section we give a method for calculating the coefficients for this linear combination. 2 The Newton-Cotes Closed Formula We wish to estimate the following integral. Z b a f ( x ) dx 1
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We use the value of the function at the following points { a + i · b - a n i = 0 , 1 , 2 , . . . , n } . Our estimate will have the following form. Z b a f ( x ) dx = A 0 · f ( a ) + A 1 · f a + b - a n + A 2 · f a + 2 · b - a n + · · · + A n · f ( b ) So, what values should we use for the coefficients and how can we calculate them?
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