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# View of /branches/UGA/7.4.3.pg

Sat Jul 24 17:11:33 2010 UTC (2 years, 9 months ago) by ted shifrin
File size: 1279 byte(s)
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    1 ## DESCRIPTION
2 ## Multivariable Calculus
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS('calculus','cylindrical coordinates', 'spherical coordinates')
6
7
8 ## DBsubject('Calculus')
9 ## DBchapter('Multiple Integrals')
10 ## DBsection('Applications')
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"
24            );
25
26
27 $showPartialCorrectAnswers = 1; 28 29 Context("Numeric")->variables->are(x=>'Real',y=>'Real',z=>'Real'); 30 31$A = random(1,4);
32 $k = random(2,4); 33$B = $k*$A**2;
34 $r = sqrt($k-1)*$A; 35 36$zbar = 3*($A**2-$B)**2/(4*($A**3-3*$A*$B+2*$B**(3/2)));
37 $I = 2*pi*(8*$B**(5/2)-15*$A*$B**2+10*$A**3*$B-3*$A**5)/60; 38 39 40 TEXT(beginproblem()); 41 42 43 Context()->texStrings; 44 BEGIN_TEXT 45 46 Let $$\Omega\subset\mathbb R^3$$ be the region lying above the plane $$z = A$$ and below the sphere $$x^2+y^2+z^2=B$$. Assume its density is $$\delta = 1$$. 47 48$PAR
49 Find the $$z$$-coordinate of the center of mass of $$\Omega$$.
50 $BR 51 \{ans_rule(10)\} 52$PAR
53 Find the moment of inertia of $$\Omega$$ about the $$z$$-axis.
54 $BR 55 \{ans_rule(10)\} 56 END_TEXT 57 58 ANS(num_cmp($zbar),num_cmp(\$I));
59
60 ENDDOCUMENT();
61
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