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# Annotation of /trunk/NationalProblemLibrary/Rochester/setIntegrals0Theory/csuf_in_0_1.pg

 1 : jjholt 197 ## DESCRIPTION 2 : ## Calculus 3 : ## ENDDESCRIPTION 4 : jj 144 5 : jjholt 197 ## KEYWORDS('integral' 'summation' 'limit') 6 : ## Tagged by tda2d 7 : 8 : ## DBsubject('Calculus') 9 : ## DBchapter('Techniques of Integration') 10 : jjholt 483 ## DBsection('Approximating Integrals') 11 : jjholt 197 ## Date('') 12 : ## Author('') 13 : ## Institution('Rochester') 14 : ## TitleText1('') 15 : ## EditionText1('') 16 : ## AuthorText1('') 17 : ## Section1('') 18 : ## Problem1('') 19 : 20 : jj 144 DOCUMENT(); # This should be the first executable line in the problem. 21 : 22 : loadMacros( 23 : "PG.pl", 24 : "PGbasicmacros.pl", 25 : "PGchoicemacros.pl", 26 : "PGanswermacros.pl", 27 : "PGauxiliaryFunctions.pl" 28 : ); 29 : 30 : TEXT(beginproblem()); 31 : $showPartialCorrectAnswers = 1; 32 : 33 :$a = random(1,4,2); 34 : $a2 =$a*2; 35 : $b = random(2,9,1)*random(-1,1,2); 36 :$c = random(3,5,1); 37 : 38 : BEGIN_TEXT 39 : 40 : The interval $$[0,c]$$ is partitioned into $$n$$ equal subintervals, and a number $$x_i$$ is arbitrarily chosen in the $$i^{th}$$ subinterval for each $$i$$. Then 41 : $BR 42 : $$\displaystyle \lim_{n\to \infty} \sum_{i=1}^{n}\frac{a2 x_i + b}{n} =$$ \{ans_rule(30)\} 43 : END_TEXT 44 : 45 :$answer = $a*$c + $b; 46 : 47 : ANS(num_cmp($answer)); 48 : 49 : SOLUTION(EV3(<<'EOF')); 50 : $SOL$BR 51 : Let's interpret the sum as a Riemann sum. 52 : $BR Recall that the Riemann sum 53 : for a function $$f(x)$$ on the interval $$[0,c]$$ has the form 54 : $$\displaystyle \sum_{i=1}^n f(x_i) \frac{c}{n}$$ since the length of each subinterval is $$\displaystyle \Delta x = \frac{c}{n}$$. 55 :$BR $BR 56 : $$\displaystyle \sum_{i=1}^{n}\frac{a2 x_i + b}{n} = 57 : \sum_{i=1}^{n}\frac{a2 x_i + b}{c} \cdot \frac{c}{n}$$, therefore the given sum is the Riemann sum for $$\displaystyle f(x) = \frac{a2 x + b}{c}$$. 58 :$BR $BR The limit of the Riemann sum as $$n$$ approaches infinity is the integral of the function $$f(x)$$ from $$0$$ to $$c$$, thus 59 :$BR \$BR 60 : $$\displaystyle \lim_{n\to \infty} \sum_{i=1}^{n}\frac{a2 x + b}{c} \cdot \frac{c}{n} = \int_0^{c} \frac{a2 x + b}{c} dx = \frac{1}{c} \int_0^{c} (a2 x + b) dx = 61 : \left. \frac{1}{c} \left( a x^2 + b x \right) \right|_0^{c} = 62 : \frac{1}{c} \left( a \cdot c^2 + b\cdot c\right) = answer$$ 63 : 64 : EOF 65 : 66 : ENDDOCUMENT(); # This should be the last executable line in the problem. 67 :