Volume of Solids of Revolution

Volume of solids of revolution

Complete Code

POD for Macro Files

answerHints.pl

PG problem file

Explanation

DOCUMENT();

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

Preamble

Load the answerHints.pl macro to give student feedback on particular incorrect answers. Note that the niceTables.pl macro is loaded via the PGML.pl macro, and is used to construct a table that will make the answer rules for the limits of integration appear at the top and bottom of the integral symbol.

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');

$upperLimit = NEW_ANS_NAME();
$lowerLimit = NEW_ANS_NAME();
$integrand  = NEW_ANS_NAME();
$volume     = NEW_ANS_NAME();

Setup

First, the variables x, dx, y, and dy are added to the context.

Then the limits of integration, the integrand, and the volume are computed.

Finally, answer names are generated for all of the answers in the problem. Never use hard coded made up answer names in problems. Always obtain an answer name using the NEW_ANS_NAME method. Problems that use hard coded made up answer names will not work correctly in many situations. Also note that answer names are not actually needed for this problem. They could be removed entirely and it would still function correctly. However, they are needed for the method to give full credit for a correct volume answer if the answers given for the other answer rules are left blank (or are correct) that is described in the statement section.

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.

[#
    [. .] [.[_]{$upper->cmp(weight => 5)}{4}{$upperLimit}.]*

    [.[`V =`].]
    [.[``\int``].]
    [.
        [_]{
            $int->cmp(weight => 40)->withPostFilter(AnswerHints(
                Formula('pi x^2 - pi x^4 dx') =>
                    "Don't forget to multiply every term in the integrand by dx",
                Formula('pi(x^2 - x^4)')   => "Don't forget the differential dx",
                Formula('pi(x^4 - x^2)dx') => 'Is the parabola above the line?',
                Formula('pi(x^4 - x^2)')   => 'Is the parabola above the line?',
                Formula('pi(x - x^2)')     => 'Make sure you use the disk method.',
                Formula('pi(x - x^2)dx')   => 'Make sure you use the disk method.',
            ))
        }{10}{$integrand}
    .]
    [.[`\;=\;`].]
    [.[_]{$vol->cmp(weight => 50)}{4}{$volume}.]*

    [. .] [.[_]{$lower->cmp(weight => 5)}{4}{$lowerLimit}.]
#]*{
    align      => 'rl',
    valign     => 'middle',
    allcellcss => { padding => '3pt' }
}
END_PGML

Statement

Standard PGML is used to describe the problem.

Then the integral is formatted with a LayoutTable from the niceTables.pl macro using its PGML syntax. Note that each answer rule is given the appropriate name from the names generated before in the last answer rule option. In addition the weight for each answer (the percent that answer is worth in the final problem score) is passed via the weight option to the cmp method.

Answer hints for various incorrect answers are provided for the integrand answer by adding ->withPostFilter(AnswerHints(...)) to the cmp call.

If you would like to give full credit for a correct volume answer if the answers given for the other answer rules are left blank (or are correct), then change [.[_]{$vol->cmp(weight => 50)}{4}{$volume}.]* to

[.
    [_]{
        $vol->cmp(
            weight => 50,
            credit => [ $upperLimit, $lowerLimit, $integrand ]
        )
    }{4}{$volume}
.]*

If you want to give equal credit for all answers, then remove the weight options from the cmp calls.

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

A solution should be provided here.