View of /trunk/NationalProblemLibrary/ASU-topics/setCalculus/stef/stef15_7p9.pg

    1 ## DESCRIPTION
    2 ## Multivariable Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('calculus','iterated integral')
    6 ## Tagged by cmd6a 3/12/06
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Multiple Integrals')
   10 ## DBsection('Triple Integrals')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('ASU')
   14 ## TitleText1('Calculus: Early Transcendentals')
   15 ## EditionText1('5')
   16 ## AuthorText1('Stewart')
   17 ## Section1('15.7')
   18 ## Problem1('28')
   19 
   20 DOCUMENT();
   21 loadMacros("PG.pl",
   22            "PGbasicmacros.pl",
   23            "PGchoicemacros.pl",
   24            "PGanswermacros.pl",
   25            "PGauxiliaryFunctions.pl",
   26            "PGgraphmacros.pl",
   27            "Dartmouthmacros.pl");
   28 
   29 
   30 ## Do NOT show partial correct answers
   31 $showPartialCorrectAnswers = 1;
   32 
   33 $a = random(1,8);
   34 $b = random(1,8);
   35 $aa = $a**2;
   36 $p = nicestring([$b],["y"]);
   37 
   38 TEXT(beginproblem());
   39 
   40 
   41 BEGIN_TEXT
   42 $BR
   43 Express the integral
   44 \(\displaystyle \iiint_E f(x,y,z) dV \) as an iterated integral
   45 in six different ways, where E is the solid bounded by
   46 \( z =0,  z = $p \) and \( x^2 = $aa -y \).
   47  $PAR
   48 1.
   49 \(\displaystyle \int_a^b
   50 \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dy dx \)
   51 
   52 $BR
   53     \( a = \) \{ans_rule()\}   \( b = \) \{ans_rule()\}
   54 
   55 $BR
   56     \( g_1(x) = \) \{ans_rule()\}   \( g_2(x) = \) \{ans_rule()\}
   57 
   58 $BR
   59     \( h_1(x,y) = \) \{ans_rule()\}   \( h_2(x,y) = \) \{ans_rule()\}
   60 $PAR
   61 
   62 2.
   63 \(\displaystyle \int_a^b
   64 \int_{g_1(y)}^{g_2(y)} \int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dx dy \)
   65 
   66 $BR
   67     \( a = \) \{ans_rule()\}   \( b = \) \{ans_rule()\}
   68 
   69 $BR
   70     \( g_1(y) = \) \{ans_rule()\}   \( g_2(y) = \) \{ans_rule()\}
   71 
   72 $BR
   73     \( h_1(x,y) = \) \{ans_rule()\}   \( h_2(x,y) = \) \{ans_rule()\}
   74 $PAR
   75 
   76 3.
   77 \(\displaystyle \int_a^b
   78 \int_{g_1(z)}^{g_2(z)} \int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dy dz \)
   79 
   80 $BR
   81     \( a = \) \{ans_rule()\}   \( b = \) \{ans_rule()\}
   82 
   83 $BR
   84     \( g_1(z) = \) \{ans_rule()\}   \( g_2(z) = \) \{ans_rule()\}
   85 
   86 $BR
   87     \( h_1(y,z) = \) \{ans_rule()\}   \( h_2(y,z) = \) \{ans_rule()\}
   88 $PAR
   89 
   90 4.
   91 \(\displaystyle \int_a^b
   92 \int_{g_1(y)}^{g_2(y)} \int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dz dy \)
   93 
   94 $BR
   95     \( a = \) \{ans_rule()\}   \( b = \) \{ans_rule()\}
   96 
   97 $BR
   98     \( g_1(y) = \) \{ans_rule()\}   \( g_2(y) = \) \{ans_rule()\}
   99 
  100 $BR
  101     \( h_1(y,z) = \) \{ans_rule()\}   \( h_2(y,z) = \) \{ans_rule()\}
  102 $PAR
  103 
  104 5.
  105 \(\displaystyle \int_a^b
  106 \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dz dx \)
  107 
  108 $BR
  109     \( a = \) \{ans_rule()\}   \( b = \) \{ans_rule()\}
  110 
  111 $BR
  112     \( g_1(x) = \) \{ans_rule()\}   \( g_2(x) = \) \{ans_rule()\}
  113 
  114 $BR
  115     \( h_1(x,z) = \) \{ans_rule()\}   \( h_2(x,z) = \) \{ans_rule()\}
  116 $PAR
  117 
  118 6.
  119 \(\displaystyle \int_a^b
  120 \int_{g_1(z)}^{g_2(z)} \int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dx dz \)
  121 
  122 $BR
  123     \( a = \) \{ans_rule()\}   \( b = \) \{ans_rule()\}
  124 
  125 $BR
  126     \( g_1(z) = \) \{ans_rule()\}   \( g_2(z) = \) \{ans_rule()\}
  127 
  128 $BR
  129     \( h_1(x,z) = \) \{ans_rule()\}   \( h_2(x,z) = \) \{ans_rule()\}
  130 $PAR
  131 
  132 END_TEXT
  133 
  134 ### 1. ###
  135 ANS(num_cmp(-$a)); ANS(num_cmp($a));
  136 ANS(fun_cmp("0", vars=>"x")); ANS(fun_cmp("$aa - x**2", vars=>"x"));
  137 ANS(fun_cmp("0", vars=>["y","x"]));
  138 ANS(fun_cmp("$b*y", vars=>["y","x"]));
  139 
  140 ### 2. ####
  141 ANS(num_cmp(0)); ANS(num_cmp($aa));
  142 ANS(fun_cmp("-sqrt($aa-y)", vars=>"y")); ANS(fun_cmp("sqrt($aa-y)", vars=>"y"));
  143 ANS(fun_cmp("0", vars=>["y","x"]));
  144 ANS(fun_cmp("$b*y", vars=>["y","x"]));
  145 
  146 ####  3. ####
  147 ANS(num_cmp(0)); ANS(num_cmp($b*$aa));
  148 ANS(fun_cmp("z/$b", vars=>"z")); ANS(fun_cmp("$aa", vars=>"z"));
  149 ANS(fun_cmp("-sqrt($aa-y)", vars=>["y","z"]));
  150 ANS(fun_cmp("sqrt($aa-y)", vars=>["y","z"]));
  151 
  152 
  153 #### 4. ####
  154 ANS(num_cmp(0)); ANS(num_cmp($aa));
  155 ANS(fun_cmp("0", vars=>"y")); ANS(fun_cmp("$b*y", vars=>"y"));
  156 ANS(fun_cmp("-sqrt($aa-y)", vars=>["y","z"]));
  157 ANS(fun_cmp("sqrt($aa-y)", vars=>["y","z"]));
  158 
  159 
  160 #####  5. #####
  161 ANS(num_cmp(-$a)); ANS(num_cmp($a));
  162 ANS(fun_cmp("0", vars=>"x")); ANS(fun_cmp("$b*($aa-x^2)", vars=>"x"));
  163 ANS(fun_cmp("z/$b", vars=>["x","z"]));
  164 ANS(fun_cmp("$aa - x^2", vars=>["x","z"]));
  165 
  166 ##### 6. #####
  167 ANS(num_cmp(0)); ANS(num_cmp($b*$aa));
  168 ANS(fun_cmp("-sqrt($aa-z/$b)", vars=>"z")); ANS(fun_cmp("sqrt($aa-z/$b)", vars=>"z"));
  169 ANS(fun_cmp("z/$b", vars=>["x","z"]));
  170 ANS(fun_cmp("$aa-x^2", vars=>["x","z"]));
  171 
  172 
  173 
  174 
  175 ENDDOCUMENT();