DoubleIntegral1

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This article has been retained as a historical document. It is not up-to-date and the formatting may be lacking. Use the information herein with caution.

This problem has been replaced with a newer version of this problem

Setting up a Double Integral

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This PG code shows how to allow students to set up a double integral and integrate in either order.


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PG problem file Explanation

Problem tagging data

Problem tagging: 

DOCUMENT();        

loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"parserMultiAnswer.pl",
);

TEXT(beginproblem());

Initialization: Since there are multiple answer blanks that are dependent upon each other, we use parserMultiAnswer.pl. 

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;
		}
	}
);

Setup: There are two separate cases: integrating with respect to dx dy (which we call case 0) or with respect to dy dx (which we call case 1). The zeroth and first entries in each of the arrays @id, @od, @A, @B, @C, @D hold the values for case 0 and case 1, respectively. We used constant limits of integration to keep this example easy to follow, but we encourage you 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 ``]
[___________]{$multians} [_____]{$multians} [_____]{$multians}

A = [_____________]{$multians}
B = [_____________]{$multians}
C = [_____________]{$multians}
D = [_____________]{$multians}

Volume = [___________________________]{$V}

Main Text: The only interesting thing to note here is that you must use $multians for each answer blank (except the last one, which is independent.) 

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

COMMENT('Allows integration in either order.  Uses PGML.');
ENDDOCUMENT();

Solution:

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