## DESCRIPTION ## Calculus ## ENDDESCRIPTION ## KEYWORDS('integral' 'summation' 'limit') ## Tagged by tda2d ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Approximating Integrals') ## Date('') ## Author('') ## Institution('Rochester') ## TitleText1('') ## EditionText1('') ## AuthorText1('') ## Section1('') ## Problem1('') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl", "PGcourse.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a = random(1,4,2); $a2 =$a*2; $b = random(2,9,1)*random(-1,1,2);$c = random(3,5,1); BEGIN_TEXT 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 $BR $$\displaystyle \lim_{n\to \infty} \sum_{i=1}^{n}\frac{a2 x_i + b}{n} =$$ \{ans_rule(30)\} END_TEXT$answer = $a*$c + $b; ANS(num_cmp($answer)); SOLUTION(EV3(<<'EOF')); $SOL$BR Let's interpret the sum as a Riemann sum. $BR Recall that the Riemann sum for a function $$f(x)$$ on the interval $$[0,c]$$ has the form $$\displaystyle \sum_{i=1}^n f(x_i) \frac{c}{n}$$ since the length of each subinterval is $$\displaystyle \Delta x = \frac{c}{n}$$.$BR $BR $$\displaystyle \sum_{i=1}^{n}\frac{a2 x_i + b}{n} = \sum_{i=1}^{n}\frac{a2 x_i + b}{c} \cdot \frac{c}{n}$$, therefore the given sum is the Riemann sum for $$\displaystyle f(x) = \frac{a2 x + b}{c}$$.$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$BR \$BR $$\displaystyle \lim_{n\to \infty} \sum_{i=1}^{n}\frac{a2 x + b}{c} \cdot \frac{c}{n} = \int_0^{c} \frac{a2 x + b}{c} dx = \frac{1}{c} \int_0^{c} (a2 x + b) dx = \left. \frac{1}{c} \left( a x^2 + b x \right) \right|_0^{c} = \frac{1}{c} \left( a \cdot c^2 + b\cdot c\right) = answer$$ EOF ENDDOCUMENT(); # This should be the last executable line in the problem.