## DESCRIPTION
## Multivariable Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus','change of variables')


## DBsubject('Calculus')
## DBchapter('Multiple Integrals')
## DBsection('Change of Variables')
## Date('December 28, 2009')
## Author('Ted Shifrin')
## Institution('UGA')
## TitleText1()


DOCUMENT();
loadMacros("PG.pl",
           "PGbasicmacros.pl",
           "PGchoicemacros.pl",
           "PGanswermacros.pl",
           "PGauxiliaryFunctions.pl",
           "PGmatrixmacros.pl",
           "Parser.pl",
           );
 

$showPartialCorrectAnswers = 1;

Context("Numeric")->variables->are(x=>'Real',y=>'Real');

$a = random(2,4);
$b = random($a,5);
$A = $a**2;


@f = ("x^2","y^2","(y/x)^2", "(y/x)^3");
@choice = NchooseK($#f,1);
$f = Formula("@f[@choice[0]]");

@ans = ("(2-2*$a^6-3*$b^2+3*$a^4*$b^2)/(12*$b^2)","$b^2*($a^2-1)/2+(1-$a^4)/4","($b^4-$a^2)*(1-1/$a^2)/4","($b^6(1-1/$a^4)+2*(1-$a^2))/12");
$ans = @ans[@choice[0]];

TEXT(beginproblem());

Context()->texStrings;
BEGIN_TEXT

Let \(S\subset\mathbb R^2\) be the region in the first quadrant bounded by the curves \(y=x\), \(y=$b\), \(xy=1\), and \(xy=$A\). Use an appropriate change of variables to evaluate
\[ \int_S $f\,dA \ .\]
$PAR

\{ans_rule(15)\}
END_TEXT

ANS(num_cmp($ans));

ENDDOCUMENT();