## 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();