View of /branches/UGA/7.3.8.pg

    1 ## DESCRIPTION
    2 ## Multivariable Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('calculus','cylindrical coordinates')
    6 
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Multiple Integrals')
   10 ## DBsection('Cylindrical Coordinates')
   11 ## Date('December 26, 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 
   28 $showPartialCorrectAnswers = 1;
   29 
   30 Context("Numeric")->variables->are(x=>'Real',y=>'Real',z=>'Real',r=>'Real',t=>'Real',R=>'Real',phi=>'Real');
   31 
   32 $A = random(1,4);
   33 $p = Formula("$A z")->reduce;
   34 $k = random(1,4);
   35 $B = $k*($k+$A);
   36 $r = sqrt($k*$A);
   37 
   38 @fn = ("1","z","sqrt(x^2+y^2)*z");
   39 @g = ("1","z","r*z");
   40 
   41 @answers = ("pi*$k/6*(-3*$A*$k-4*$k**2+sqrt($B)*4*($A+$k))","pi*$A*$k**2*(4*$k+3*$A)/12","2*pi*(7*$A+10*$k)*$A**(3/2)*$k**(5/2)/105");
   42 
   43 @choice = NchooseK($#fn,1);
   44 $f = @fn[@choice[0]];
   45 $g = Formula("@g[@choice[0]]*r")->reduce;
   46 
   47 $ans = @answers[@choice[0]];
   48 
   49 
   50 $a = Compute("0");
   51 $b = Compute("2*pi");
   52 $C = Formula("0");
   53 $D = Formula("$r");
   54 $E = Formula("r^2/$A");
   55 $F = Formula("sqrt($B-r^2)")->reduce;
   56 
   57 
   58 
   59 
   60 TEXT(beginproblem());
   61 
   62 install_weighted_grader();
   63 Context()->texStrings;
   64 BEGIN_TEXT
   65 
   66 Let \(\Omega\subset\mathbb R^3\) be the region lying above the paraboloid \($p = x^2+y^2\) and below the sphere \(x^2+y^2+z^2=$B\). Express the triple integral
   67 
   68 \[ \int_{\Omega} $f\,dV\]
   69 
   70 as an iterated integral in cylindrical coordinates
   71 
   72 \[ \int_a^b\int_C^D\int_E^F g\left(\begin{array}{c} r\\t\\z \end{array}\right) \,dz\,dr\,dt\ ,\]
   73 
   74 $BR
   75 where
   76 $BR
   77 \(a = \) \{ans_rule(5)\}, \(b = \) \{ans_rule(5)\},
   78 $BR
   79 \(C = \) \{ans_rule(8)\},  \(D = \) \{ans_rule(8)\},
   80 $BR
   81 \(E = \) \{ans_rule(8)\},  \(F = \) \{ans_rule(8)\}, and
   82 $PAR
   83 \(g\left(\begin{array}{c} r\\t\\z \end{array}\right) = \) \{ans_rule(12)\}.
   84 
   85 $PAR
   86 $BBOLD
   87 Note: $EBOLD We are using \(t\) for the usual \(\theta\), so that you don't have to type "theta" each time in WeBWork.
   88 $PAR
   89 
   90 
   91 Now evaluate your iterated integral to compute the original triple integral.
   92 $PAR
   93 \{ans_rule(20)\}
   94 
   95 END_TEXT
   96 
   97 WEIGHTED_ANS($a->cmp,5,$b->cmp,5);
   98 WEIGHTED_ANS($C->cmp,5,$D->cmp,15);
   99 WEIGHTED_ANS($E->cmp,10,$F->cmp,10);
  100 WEIGHTED_ANS($g->cmp,10);
  101 
  102 WEIGHTED_ANS(num_cmp($ans),40);
  103 
  104 ENDDOCUMENT();
  105 
  106