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

Wed Jul 18 01:12:07 2007 UTC (5 years, 11 months ago) by jjholt
File size: 4186 byte(s)
Fixed title, edition tags.

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();
22            "PGbasicmacros.pl",
23            "PGchoicemacros.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();