## 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();