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| 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(); |
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