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# View of /branches/UGA/7.3.8.pg

Sat Jul 24 17:11:33 2010 UTC (2 years, 10 months ago) by ted shifrin
File size: 2375 byte(s)
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    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();
19            "PGbasicmacros.pl",
20            "PGchoicemacros.pl",
22            "PGauxiliaryFunctions.pl",
23            "Parser.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
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