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# Annotation of /branches/UGA/7.2.8.pg

 1 : ted shifri 1458 ## 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();