View of /trunk/NationalProblemLibrary/Rochester/setIntegrals0Theory/ur_in_0_12.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('integral' 'summation' 'area' 'riemann')
    6 ## Tagged by tda2d
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Integrals')
   10 ## DBsection('Area and Distance')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 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 $an = random(2,4,1);
   34 $as = random(-1,1,2);
   35 if ($as == 1) { $s = ' ' }
   36 if ($as == -1) { $s = '-'}
   37 $a  = $an * $as;
   38 
   39 $bn = random(1,9,1);
   40 $bs = random(-1,1,2);
   41 while ($bn == $an) {
   42   $bn = random(1,9,1);
   43   }
   44 $b  = $bn * $bs;
   45 
   46 $c  = random(2,4,1);
   47 
   48 BEGIN_TEXT
   49 Consider the function \( \displaystyle f(x) = $s \frac {x^2}{$an} + $b \).
   50 $PAR
   51 In this problem you will calculate
   52 \( \displaystyle \int_{0}^{$c} \left( $s \frac {x^2}{$an} + $b \right) \,dx \)
   53 by using the
   54 definition \[ \int_{a}^{b} f(x) \,dx = \lim_{n \to \infty} \left[ \sum_{i=1}^{n} f(x_i) \Delta x \right] \]
   55 $PAR
   56 The summation inside the brackets is \( R_n \) which is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval.
   57 $PAR
   58 Calculate \( R_n \) for \( \displaystyle f(x) = $s \frac {x^2}{$an} + $b \) on the interval \( [0, $c] \) and write your answer as a function of \( n \) without any summation signs.  You will need the summation formulas on page 381 of your textbook
   59 (page 364 in older texts).
   60 END_TEXT
   61 
   62 HINT(EV2(<<EOT));
   63 \( \displaystyle x_i = \frac {$c i} {n} \) and  \( \displaystyle \Delta x = \frac {$c} {n} \) .
   64 EOT
   65 
   66 BEGIN_TEXT
   67 $BR
   68 \( R_n = \) \{ans_rule(45)\}
   69 $BR
   70 \( \displaystyle \lim_{n \to \infty} R_n = \) \{ans_rule(15)\}
   71 $BR
   72 
   73 END_TEXT
   74 
   75 $ans1 = "$b*$c + $c**3*(n+1)*(2*n+1)/(6*($a)*n**2)";
   76 $ans2 = $c**3/(($a)*3) + $b*$c;
   77 
   78 ANS(fun_cmp($ans1, vars=>'n'));
   79 ANS(num_cmp($ans2));
   80 
   81 ENDDOCUMENT();