## DESCRIPTION ## Multivariable integral calculus: setting up double integrals ## ENDDESCRIPTION ## KEYWORDS('Integrals', 'setting up double integrals') ## DBsubject('WeBWorK') ## DBchapter('WeBWorK Tutorial') ## DBsection('Fort Lewis Tutorial 2011') ## Date('10/20/2010') ## Author('Paul Pearson') ## Institution('Fort Lewis College') ## TitleText1('') ## EditionText1('') ## AuthorText1('') ## Section1('') ## Problem1('') ################################## # Initialization DOCUMENT(); loadMacros( "PGstandard.pl", "MathObjects.pl", "parserMultiAnswer.pl", ); TEXT(beginproblem()); ################################### # Setup Context("Numeric"); Context()->variables->are( x=>"Real",dx=>"Real", y=>"Real",dy=>"Real"); Context()->flags->set(reduceConstants=>0); # # limits of integration # $a = random(1,5,1);$b = $a + random(1,4,1); do {$c = random(1,5,1); } until ($c !=$a); do { $d =$c + random(1,4,1); } until ($d !=$b); # # integrand and volume # $f = Formula("x*y");$V = Formula("($b^2-$a^2) * ($d^2-$c^2) / 4"); # # differentials and limits of integration # # Case 0, element 0 of each array below, is # if the order of integration is dx dy # # Case 1, element 1 of each array below, is # if the order of integration is dy dx # # "id" and "od" stand for inner and outer differential # @id = (Formula("dx"),Formula("dy")); # (case 0, case 1) @od = (Formula("dy"),Formula("dx")); # (case 0, case 1) # # A = outer integral, lower limit # B = outer integral, upper limit # C = inner integral, lower limit # D = inner integral, upper limit # @A = (Formula("$c"),Formula("$a")); # (case 0, case 1) @B = (Formula("$d"),Formula("$b")); # (case 0, case 1) @C = (Formula("$a"),Formula("$c")); # (case 0, case 1) @D = (Formula("$b"),Formula("$d")); # (case 0, case 1) $multians = MultiAnswer($f, $id[0],$od[0], $A[0],$B[0], $C[0],$D[0] )->with( singleResult => 1, checker => sub { my ( $correct,$student, $self ) = @_; my ($fstu, $idstu,$odstu, $Astu,$Bstu, $Cstu,$Dstu ) = @{$student}; if ( ($f == $fstu &&$id[0] == $idstu &&$od[0] == $odstu &&$A[0] == $Astu &&$B[0] == $Bstu &&$C[0] == $Cstu &&$D[0] == $Dstu ) || ($f == $fstu &&$id[1] == $idstu &&$od[1] == $odstu &&$A[1] == $Astu &&$B[1] == $Bstu &&$C[1] == $Cstu &&$D[1] == $Dstu ) ) { return 1; } elsif ( ($f == $fstu &&$id[0] == $idstu &&$od[0] == $odstu && ($A[0] != $Astu ||$B[0] != $Bstu) &&$C[0] == $Cstu &&$D[0] == $Dstu ) || ($f == $fstu &&$id[1] == $idstu &&$od[1] == $odstu && ($A[1] != $Astu ||$B[1] != $Bstu) &&$C[1] == $Cstu &&$D[1] == $Dstu ) || ($f == $fstu &&$id[0] == $idstu &&$od[0] == $odstu &&$A[0] == $Astu &&$B[0] == $Bstu && ($C[0] != $Cstu ||$D[0] != $Dstu) ) || ($f == $fstu &&$id[1] == $idstu &&$od[1] == $odstu &&$A[1] == $Astu &&$B[1] == $Bstu && ($C[1] != $Cstu ||$D[1] != $Dstu) ) ) {$self->setMessage(1,"Check your limits of integration."); return 0.94; } elsif ( ( $f ==$fstu && $id[0] ==$idstu && $od[0] ==$odstu && ($A[0] !=$Astu || $B[0] !=$Bstu) && ($C[0] !=$Cstu || $D[0] !=$Dstu) ) || ( $f ==$fstu && $id[1] ==$idstu && $od[1] ==$odstu && ($A[1] !=$Astu || $B[1] !=$Bstu) && ($C[1] !=$Cstu || $D[1] !=$Dstu) ) ) { $self->setMessage(1, "Check your limits of integration and order of integration."); return 0.47; } else { return 0; } } ); ##################################### # Main text Context()->texStrings; BEGIN_TEXT Set up a double integral in rectangular coordinates for calculating the volume of the solid under the graph of the function $$f(x,y) = f$$ over the region $$a \leq x \leq b$$ and $$c \leq y \leq d$$.$BR $BR${BITALIC}Instructions:${EITALIC} Please enter the integrand in the first answer box. Depending on the order of integration you choose, enter${BITALIC}dx${EITALIC} and${BITALIC}dy${EITALIC} in either order into the second and third answer boxes with only one${BITALIC}dx${EITALIC} or${BITALIC}dy${EITALIC} in each box. Then, enter the limits of integration and evaluate the integral to find the volume.$BR $BR $$\displaystyle \int_A^B \int_C^D$$ \{$multians->ans_rule(40) \} \{ $multians->ans_rule(5) \} \{$multians->ans_rule(5) \} $BR$BR A = \{ $multians->ans_rule(20) \}$BR B = \{ $multians->ans_rule(20) \}$BR C = \{ $multians->ans_rule(20) \}$BR D = \{ $multians->ans_rule(20) \}$BR $BR Volume = \{ ans_rule(40) \} END_TEXT Context()->normalStrings; #################################### # Answer Evaluation$showPartialCorrectAnswers = 1; ANS( $multians->cmp() ); ANS($V->cmp() ); #################################### # Solution Context()->texStrings; BEGIN_SOLUTION ${PAR}SOLUTION:${PAR} Solution explanation goes here. END_SOLUTION Context()->normalStrings; COMMENT('MathObject version. Allows integration in either order.'); ENDDOCUMENT();