HW11suggestions

# HW11suggestions - points 11.3 Give an approximate...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 11 120202: ESM4A - Numerical Methods Homework Problems 11.1. Calculate the following integrals to seven-decimal accuracy (a) Z 1 0 e x 2 dx (b) Z 0 . 01 0 ± sin x x ² dx ( ? points ) 11.2. (a) Find the coeﬃcients and nodes of a Gaussian quadrature formula of the form Z 1 0 x 4 f ( x ) dx A 0 f ( x 0 ) + A 1 f ( x 1 ) (b) Derive a two-point integration formula for integrals of the form R 1 - 1 f ( x )(1+ x 2 ) dx , which is exact when f ( x ) is a polynomial of degree 3. (c) Apply the result of (b) to f ( x ) = x 4 . Use the result to derive a reminder term.
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Unformatted text preview: ( ? points ) 11.3. Give an approximate evaluation, by the Monte Carlo method, of the integral Z Z ( x 2 + y 2 ) dx dy where the domain of integration is deﬁned by the inequalities 1 2 ≤ x ≤ 1 , ≤ y ≤ 2 x-1. Remark : Take N = 20 random points; for the sake of simplicity, you can round their coordinates to three decimal places. ( ? points ) Due: 7.05.10, at 3 pm (in the mailbox labeled Linsen in the entrance hall of Res.I )...
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