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Lecture16wn_NucE309FA11

Lecture16wn_NucE309FA11 - NucE 309 Fall 2011 Lecture 16...

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16.1 NucE 309, Fall 2011, Lecture 16 19.5 Numerical Integration: Pages 817-818 (was 17.5 pages 869-870) We would like to approximate the value of the integral ( ) b a J f x dx by dividing the range into segments and approximating f(x) dx over each segment. Divide the range into n segments. The width of each segment is: b a h n A. Rectangular Rule. Approximate f(x) dx using the value of f(x) at the midpoint of each segment. 1/ 2 3/ 2 5/ 2 1/ 2 ( ) ( ) ( ) ( ) ( ) b n a J f x dx h f x f x f x f x B. Trapezoidal Rule. Approximate f(x) dx using the area of a trapezoid between x i and x i+1 . The area of that trapezoid is 1 ( ) ( ) 2 i i h dA f x f x . Add up all of these dA to give 1 1 2 2 3 3 2 2 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 b a n n n n n J f x dx h f a f x f x f x f x f x f x f x f x f x f x f b simplifies to: 1 2 1 1 ( ) ( ) ( ) ( ) ( ) 2 2 b a J f x dx h f a f x f x f b
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