Volume of Solids of Revolution 2

Volume of solids of revolution

Complete Code

Download file: Volume2.pg

POD for Macro Files

PG problem file

Explanation

DOCUMENT();

loadMacros(
    'PGstandard.pl',     'PGML.pl',
    'unionTables.pl',    'answerHints.pl',
    'weightedGrader.pl', 'PGcourse.pl'
);

install_weighted_grader();

Preamble

We load weightedGrader.pl and install it. We load answerHints.pl to give student feedback on particular incorrect answers. We load unionTables.pl so that we can construct tables in HTML mode that will make the answer blanks for the limits of integration appear at the top and bottom of the integral symbol.

If the weighted grader is to be used, the command install_weighted_grader(); must be called.

Context()->variables->are(x => 'Real', dx => 'Real', y => 'Real', dy => 'Real');

$f = Compute('x');
$g = Compute('x^2');

$upper = Real('1');
$lower = Real('0');
$int   = Compute('(pi x - pi x^2) dx');
$vol   = Compute('pi');

@weights = (5, 5, 40, 50);

#  Display the answer blanks properly in different modes
if ($displayMode eq 'TeX') {
    $integral =
        'Volume = \(\displaystyle'
        . '\int_{'
        . NAMED_ANS_RULE('lowerlimit', 4) . '}^{'
        . NAMED_ANS_RULE('upperlimit', 4) . '}'
        . NAMED_ANS_RULE('integrand',  30) . ' = '
        . ans_rule(10) . '\)';
} else {
    $integral = BeginTable(center => 0)
        . Row(
            [
                'Volume = \(\displaystyle\int\)',
                NAMED_ANS_RULE('upperlimit', 4)
                . $BR
                . $BR
                . NAMED_ANS_RULE('lowerlimit', 4),
                NAMED_ANS_RULE('integrand', 30)
                . $SPACE . ' = '
                . $SPACE
                . ans_rule(10),
            ],
            separation => 2
        ) . EndTable();
}

Setup

Notice that for the final answer (volume) we use ans_rule(width), while for the answer blanks that involve setting up the integral we use NAMED_ANS_RULE(name, width).

BEGIN_PGML
Set up and evaluate an integral for the volume of the solid of revolution
obtained by rotating the region bounded by [`y = [$f]`] and [`y = [$g]`] about
the [`x`] -axis.

[$integral]*

[@ MODES(
    TeX  => '',
    HTML => << "END_HTML"
${BITALIC}${BBOLD}Note:${EBOLD} You can earn
$weights[0]${PERCENT} for the upper limit of integration,
$weights[1]${PERCENT} for the lower limit of integration,
$weights[2]${PERCENT} for the integrand, and
$weights[3]${PERCENT} for the finding the volume.
If you find the correct volume, you will get full credit
no matter what your other answers are.
${EITALIC}
END_HTML
) @]*
END_PGML

Statement

This is the problem statement in PGML.

#; which can provide full credit, we use
NAMED_WEIGHTED_ANS(upperlimit => $upper->cmp, $weights[0]);
NAMED_WEIGHTED_ANS(lowerlimit => $lower->cmp, $weights[1]);
NAMED_WEIGHTED_ANS(
    integrand => $int->cmp->withPostFilter(AnswerHints(
        Formula('pi x - pi x^2 dx') =>
            "Don't forget to multiply every term in the integrand by dx",
        Formula('pi x - pi x^2')      => "Don't forget the differential dx",
        Formula('(pi x^2 - pi x)*dx') => 'Is the parabola above the line?',
        Formula('pi x^2 - pi x')      => 'Is the parabola above the line?',
    )),
    $weights[2]
);
CREDIT_ANS($vol->cmp, [ 'upperlimit', 'lowerlimit', 'integrand' ], $weights[3]);

Answer

Notice that we use NAMED_WEIGHTED_ANS(name => $answer->cmp()->withPostFilter(), weight) for the questions that have named answer blanks above. For the final answer, CREDIT_ANS( $answer->cmp(), ['name1', 'name2', ...], weight) for this answer to provide credit for the answers with names in the list ['name1', 'name2', ...].

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

COMMENT(
    'Weights each answer blank separately, and the last answer provides full '
        . 'credit for all other answer blanks.');

ENDDOCUMENT();

Solution

A solution should be provided here.