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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 ) = ex 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. ex 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+(i1)*(ba)/n]*(ba)/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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 Spring '11
 Johnson
 Math, Riemann integral, Riemann sum, Riemann, left riemann sums, )∆xk .

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