View of /branches/UGA/7.4.3.pg

    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();
   18 loadMacros("PG.pl",
   19            "PGbasicmacros.pl",
   20            "PGchoicemacros.pl",
   21            "PGanswermacros.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 
   62