DoubleIntegral.pg

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;
        }
    }
);

Setup

There are two separate cases: integrating with respect to dx dy (case 0) or with respect to dy dx (case 1). The zeroth and first entries in each of the arrays @id, @od, @A, @B, @C, and @D hold the values for case 0 and case 1, respectively. Constant limits of integration are used to keep this example easy to follow, but you are encouraged to write questions over non-rectangular regions.

The $multians object has been compartmentalized, so you shouldn’t need to change it unless you want to fiddle with the weighted score for each answer blank (by changing the return values). The return values are set so that the percentages come out nicely.

BEGIN_PGML
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]`].

_Instructions:_ Please enter the integrand in the first answer box . Depending
on the order of integration you choose, enter [`dx`] and [`dy`] in either order
into the second and third answer boxes with only one [`dx`] or [`dy`] in each
box . Then, enter the limits of integration and evaluate the integral to find
the volume.

[``\int_A^B \int_C^D``] [_]{$multians}{10} [_]{$multians}{5} [_]{$multians}{5}

A = [_]{$multians}{10}

B = [_]{$multians}{10}

C = [_]{$multians}{10}

D = [_]{$multians}{10}

Volume = [_]{$V}{10}
END_PGML

Statement

The only interesting thing to note here is that $multians must be used for each answer rule (except the last one, which is independent.)

Double Integral

Complete Code

POD for Macro Files

PG problem file

Explanation