Volume of Solids of Revolution

Volume of solids of revolution

Complete Code

POD for Macro Files

PG problem file

Explanation

DOCUMENT();

loadMacros(
    'PGstandard.pl',     'PGML.pl',
    'niceTables.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 niceTables.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');

$upper = Real('1');
$lower = Real('0');
$int   = Compute('(pi x^2 - pi x^4) dx');
$vol   = Compute('2pi/15');

$integral = LayoutTable(
    [
        [ ' ', NAMED_ANS_RULE('upperlimit', 4) ],
        [
            '\(V= \)',
            '\(\displaystyle \int \;\;\)'
                . NAMED_ANS_RULE('integrand', 10)
                . '\(\;=\;  \)'
                . NAMED_ANS_RULE('volume', 4)
        ],
        [ ' ', NAMED_ANS_RULE('lowerlimit', 4) ],
    ],
    align      => 'rl',
    valign     => 'middle',
    allcellcss => { padding => '3pt' }
);

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

Setup

To keep the code that needs to be modified compartmentalized, we define the functions involved, the limits of integration, the integrand, the volume, and an array of weights (which sum to 100) for each of these answers.

The code for correctly displaying the answer blanks creates $integral which will be displayed correctly both in TeX and HTML modes. Notice that it uses NAMED_ANS_RULE(name, width) for all of the answer blanks instead of ans_rule(width).

We define $integral to display the integral and answer blanks correctly using a LayoutTable from niceTables.pl.

BEGIN_PGML
Set up and evaluate an integral for the volume of the solid of revolution
obtained by rotating the region bounded by [`y = x`] and [`y = x^2`] 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.
${EITALIC}
END_HTML
) @]*
END_PGML

Statement

Standard PGML and latex is used to describe the problem. The integral that was formatted using a table above is inserted with [$integral]*

A note is added that specifies for the students how the answer will be graded (the weightedGrader.pl macro does not do this automatically, as some other graders do.)

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^2 - pi x^4 dx') =>
            "Don't forget to multiply every term in the integrand by dx",
        Formula('pi (x^2 - x^4)')  => "Don't forget the differential dx",
        Formula('pi(x^4 - x^2)dx') => 'Is the parabola above the line?',
        Formula('pi(x^4 - x^2)')   => 'Is the parabola above the line?',
        Formula('pi(x - x^2)')     => 'Make sure you use the disk method.',
        Formula('pi(x - x^2)dx')   => 'Make sure you use the disk method.',
    )),
    $weights[2]
);
NAMED_WEIGHTED_ANS('volume' => $vol->cmp, $weights[3]);

Answer

To install the answer evaluator call NAMED_WEIGHTED_ANS(name => $answer->cmp()->withPostFilter(), weight) instead of using ANS($answer->cmp()->withPostFilter()). Providing customized answer hints for students is a good idea, because the whole point of this homework exercise it to learn how to set up this integral using proper notation. If we just wanted to ask for the volume, we could have done it using only one answer blank.

If you would like to give full credit for the overall volume, you can replace the last NAMED_WEIGHTED_ANS with

CREDIT_ANS($vol->cmp, [ 'upperlimit', 'lowerlimit', 'integrand' ], $weights[3]);

Of course, if you want to give equal credit, then each of the NAMED_WEIGHTED_ANS commands can be replaced with ANS, however make sure that they listed in order.

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

COMMENT('Weights each answer blank separately.');

ENDDOCUMENT();

Solution

A solution should be provided here.