View of /branches/UGA/7.2.8.pg

    1 ## DESCRIPTION
    2 ## Multivariable Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('calculus','iterated integral')
    6 
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Multiple Integrals')
   10 ## DBsection('Triple Integrals')
   11 ## Date('December 23, 2009')
   12 ## Author('Ted Shifrin')
   13 ## Institution('UGA')
   14 ## TitleText1()
   15 
   16 
   17 DOCUMENT();
   18 loadMacros("PG.pl",
   19            "PGbasicmacros.pl",
   20            "PGchoicemacros.pl",
   21            "PGanswermacros.pl",
   22            "PGauxiliaryFunctions.pl",
   23            "Parser.pl",
   24            "weightedGrader.pl");
   25 
   26 
   27 install_weighted_grader();
   28 $showPartialCorrectAnswers = 1;
   29 
   30 Context("Numeric")->variables->are(x=>'Real',y=>'Real',z=>'Real');
   31 
   32 $a = random(1,4);
   33 $b = random(1,4);
   34 $c = random(1,4);
   35 
   36 $f1 = nicestring([$a,-1],["","x"]);
   37 $f2 = nicestring([$b,$c],["x","y"]);
   38 
   39 
   40 
   41 $C = Formula("0");
   42 $D = Formula("$b x");
   43 $E = Formula("0");
   44 $F = Formula("$a-x");
   45 
   46 $DD = Formula("$b x+$c*($a-x)");
   47 $EE = Formula("(z-$b x)/$c");
   48 
   49 
   50 TEXT(beginproblem());
   51 
   52 BEGIN_TEXT
   53 $BR
   54 Assume \(f\) is continuous. Interpret the following iterated integral as a triple integral \(\int_{\Omega} f\,dV\) for the appropriate region \(\Omega\) and then change the order of integration as directed. If you do not need
   55 the second set of iterated integrals, fill in the first two limits as \(0\); if you do need both, use the first set of iterated integrals for the lower of the two regions.
   56 
   57 $PAR
   58 
   59 \[\int_0^{$a} \int_0^{$f1}\int_0^{$f2}  f\left(\begin{array}{c}x\\y\\z\end{array}\right)\,dz\, dy\, dx\ = \int_a^b\int_C^D\int_E^F f\left(\begin{array}{c}x\\y\\z\end{array}\right)\,dy\, dz\, dx + \]
   60 \[\int_{a'}^{b'}\int_{C'}^{D'}\int_{E'}^{F'} f\left(\begin{array}{c}x\\y\\z\end{array}\right)\,dy\, dz\, dx\ ,\]
   61 $BR
   62 where
   63 $BR
   64 \(a = \) \{ans_rule(5)\}, \(b = \) \{ans_rule(5)\},
   65 $BR
   66 \(C = \) \{ans_rule(10)\}, \(D = \) \{ans_rule(10)\},
   67 $BR
   68 \(E = \) \{ans_rule(10)\}, and \(F = \) \{ans_rule(10)\};
   69 $PAR
   70 and
   71 $BR
   72 \(a' = \) \{ans_rule(5)\}, \(b' = \) \{ans_rule(5)\},
   73 $BR
   74 \(C' = \) \{ans_rule(10)\}, \(D' = \) \{ans_rule(10)\},
   75 $BR
   76 \(E' = \) \{ans_rule(10)\}, and \(F' = \) \{ans_rule(10)\};
   77 
   78 END_TEXT
   79 
   80 WEIGHTED_ANS(num_cmp(0),.05,num_cmp($a),.05);
   81 WEIGHTED_ANS($C->cmp,.1); WEIGHTED_ANS($D->cmp,.1);
   82 WEIGHTED_ANS($E->cmp,.1); WEIGHTED_ANS($F->cmp,.1);
   83 
   84 WEIGHTED_ANS(num_cmp(0),.05,num_cmp($a),.05);
   85 WEIGHTED_ANS($D->cmp,.1); WEIGHTED_ANS($DD->cmp,.1);
   86 WEIGHTED_ANS($EE->cmp,.1); WEIGHTED_ANS($F->cmp,.1);
   87 
   88 
   89 ENDDOCUMENT();