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Fri Apr 28 21:06:19 2006 UTC (7 years ago) by jjholt
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    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
23 "PG.pl",
24 "PGbasicmacros.pl",
25 "PGchoicemacros.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();