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Latex-Homework2

# Latex-Homework2 - a to b is the number Z b a f x dx = lim n...

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Homework 12 Greggo Johnson November 7, 2010 Instructions Create a single article containing the following: 1. Include a title with the homework number, your name and the date on separate lines. 2. You article should start by displaying the following: Definition 0.1. Let f be continuous on [ a, b ], and let P = { x 0 , x 1 , . . . , x n } be any partition of [ a, b ]. For each k between 1 and n , let t k be an arbitrary number in [ x k - 1 , x k ]. Then the sum: f ( t 1 x 1 + f ( t 2 x 2 + · · · + f ( t n x n is called a Riemann Sum for f on [ a, b ] and is denoted n k =1 f ( t k x k . Thus: n X k =1 f ( t k x k = f ( t 1 x 1 + f ( t 2 x 2 + · · · + f ( t n x n Often we will assume the subintervals are all of equal length. That is, Δ x 1 = Δ x 2 = · · · Δ x n = b - a n . In this case, n X k =1 f ( t k x k = b - a n [ f ( t 1 ) + f ( t 2 ) + . . . + f ( t n )]

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The larger the number we assign to n , the closer the Riemann sum comes to approximating the definite integral. Under the assumption that all subintervals are of the same length, the definition of a definite integral becomes the following: Definition 0.2. Let f be continuous on [ a, b ]. The definite integral
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Unformatted text preview: a to b is the number Z b a f ( x ) dx = lim n →∞ n X k =1 f ( t k )Δ x provided that this limit exists. Example 0.3. Consider the functions f ( x ) = e-x and g ( x ) = 1 1+ x 2 on the interval [1 , 2]. Using left Riemann sums, we created the table to below to approximate R 2 1 f ( x ) dx and R 2 1 g ( x ) dx using various values of n . The last row is the exact answer. e-x 1 1+ x 2 n = 5 n = 10 n = 20 R 2 1 dx Table 1: Approximating with Left Riemann Sums 3. Fill in the table that you just created with the appropriate values rounded to 4 decimal places. It may help to use Mathematica, Matlab or Maple to accomplish this task. In Mathematica, the code LeftSum[n_]:=Sum[f[a+(i-1)*(b-a)/n]*(b-a)/n,{i,1,n}] will calculate a left Riemann sum for f [ x ] on [ a,b ]. Recall N[ ] will give you a decimal approximation in Mathematica....
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