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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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