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Unformatted text preview: Week 4 Outline Numerical integration The Midpoint Rule The Trapezoidal Rule Error estimates The Midpoint Rule The Trapezoidal Rule General formula of computing the volumes of solids Calculating the volumes of solids of revolution disk/washer method shell method Why Numerical Integration 1. It is impossible to evaluate the following integrals exactly Z 1 e x 2 d x, Z 1 1 1 + x 3 d x. 2. The function is determined from a scientific experiment through instrument readings or collected data. There may be no formula for the function. Approximating Integrals 1. Left Endpoint Rule: Z b a f ( x )d x L n := b a n n X k =1 f a + b a n ( k 1) . 2. Right endpoint Rule: Z b a f ( x )d x R n := b a n n X k =1 f a + b a n k . 3. Midpoint Rule: Z b a f ( x )d x M n := b a n n X k =1 f a + b a n ( k 1 2 ) . 1 4. Trapezoidal Rule: Z b a f ( x )d x T n := 1 2 ( L n + R n ) = b a 2 n ( f ( x ) + 2 f ( x 1 ) + + 2 f ( x n 1 ) + f ( x n )) , where x k = a + b a n k , k = 0 , 1 ...,n ....
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This note was uploaded on 12/09/2010 for the course MATH 1120 taught by Professor Gross during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 GROSS
 Calculus, Disk Method

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