## DESCRIPTION ## Multivariable Calculus ## ENDDESCRIPTION ## KEYWORDS('calculus','triple integral') ## DBsubject('Calculus') ## DBchapter('Multiple Integrals') ## DBsection('Triple Integrals') ## Date('December 23, 2009') ## Author('Ted Shifrin') ## Institution('UGA') ## TitleText1() DOCUMENT(); loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl", "Parser.pl", ); $showPartialCorrectAnswers = 1; Context("Numeric")->variables->are(x=>'Real',y=>'Real',z=>'Real');$a = random(1,5); $b = random(1,3);$f = Formula("$a y")->reduce;$g = Formula("$b**2-x^2")->reduce; @fn = ("1","y","z"); @choice = NchooseK($#fn,1); $h = Formula("@fn[@choice[0]]")->reduce; @ans=(Compute("8*$a*$b^5/15"),Compute("32*$a*$b^7/105"),Compute("16*$a^2*$b^7/105"));$ans=@ans[@choice[0]]; TEXT(beginproblem()); Context()->texStrings; BEGIN_TEXT Let $$\Omega\subset\mathbb R^3$$ be the region bounded below by the $$xy$$-plane, above by $$z=f$$, and on the sides by $$y=g$$. Evaluate the integral $\int_\Omega h\,dV \ .$ $PAR$PAR \{ans_rule(10)\} END_TEXT ANS(\$ans->cmp); ENDDOCUMENT();