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‎.gitignore‎

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*.swp
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.ai
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.svg
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.bak
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setDefs/
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‎Contrib/AgnesScott/Int_Applications/2-DiscWasher/calc07204.pg‎

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##DESCRIPTION
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## Volume of revolution of parabolic region around x-axis
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## Student must setup the integral, entering limits and integrand,
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## then give the numerical value.
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## Limits weighted 20%, integrand and answer weighted 50/30.
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## These percentages can be adjusted if desired.
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##ENDDESCRIPTION
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## DBsubject(Calculus - single variable)
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## DBchapter(Applications of integration)
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## DBsection(Volumes by disks)
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## Institution(Agnes Scott College)
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## Author(Larry Riddle)
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## Answer boxes for limits of integration Coding in PG - Nathan Wallach (CSS based formatting work and more)
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## Answer boxes for limits of integration Coding in PGML - Glenn Rice
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## https://webwork.maa.org/moodle/mod/forum/discuss.php?d=4767#p14157
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## Level(2)
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## MO(1)
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## TitleText1('APEX Calculus')
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## AuthorText1('Hartman')
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## EditionText1('4.0')
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## Section1('7.2')
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## Problem1('4')
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DOCUMENT();
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loadMacros(
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"PGstandard.pl",
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"MathObjects.pl",
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"PGML.pl",
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"PGgraphmacros.pl",
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"weightedGrader.pl",
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"contextFraction.pl",
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"answerHints.pl",
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"PGcourse.pl"
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);
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install_weighted_grader();
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TEXT(beginproblem());
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$showPartialCorrectAnswers = 1;
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$refreshCachedImages=1;
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Context("Numeric")->flags->set(
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formatStudentAnswer => parsed,
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reduceConstantFunctions => 0,);
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$a = random(4,16);
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$xlow = ($a==4 or $a==9 or $a==16) ? Real(-sqrt($a)) : Formula("-sqrt($a)");
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$xhigh = ($a==4 or $a==9 or $a==16) ? Real(sqrt($a)) : Formula("sqrt($a)");
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$s = sqrt($a);
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$f = Formula("$a-x^2");
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$integrand = Formula("pi*($a-x^2)^2");
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$ans = Compute("16/15*pi*$a^(5/2)");
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$rans = Real("16/15*pi*$a^(5/2)");
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$xmin = floor(-sqrt($a))-1;
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$xmax = ceil(sqrt($a))+1;
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$ymin = -1;
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$ymax = $a+1;
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$gr = init_graph($xmin,$ymin,$xmax,$ymax,axes=>[0,0], ticks=>[$xmax-$xmin,$ymax-$ymin], size=>[400,400]);
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$gr->lb('reset');
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$gr->new_color("lightblue", 214,230,244); # RGB
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$gr->new_color("darkblue", 100,100,255);
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add_functions($gr, "$f for x in [-$s,$s] using color:darkblue and weight:2");
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$gr->moveTo($xmin,0);
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$gr->lineTo($xmax,0,"darkblue",1);
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$gr->fillRegion([0.5,0.5,"lightblue"]);
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$i = 0; # Number the axes
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do {
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$xtick = $i;
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$labelx1 = new Label($xtick,-0.2, "$xtick",'black','center');
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$labelx2 = new Label(-$xtick,-0.2, "-$xtick",'black','center');
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if ($xtick!=0) {
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$gr->lb($labelx1);
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$gr->lb($labelx2);
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}
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$i =$i+1;
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} while ($i<($xmax-$xmin)-1);
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$i = 0;
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do {
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$ytick = $i;
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$labely = new Label(-0.2,$ytick, "$ytick",'black', 'right','middle');
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if ($ytick!=0) {$gr->lb($labely);}
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$i =$i+1;
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} while ($i<($ymax-$ymin)-1);
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# Code to format the answer boxes for integration limits
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TEXT( MODES(
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HTML=>"
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<style>
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.lowerIntegrationBoundOfPair input[type=text].codeshard {
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padding:1px;
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font-size:11pt;
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height:20px !important;
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}
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.upperIntegrationBoundOfPair input[type=text].codeshard {
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padding:1px;
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font-size:11pt;
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height:20px !important;
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}
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.divOnLineWithIntegrationLimits {
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display:inline-block;
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padding-top: 15px;
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position: relative;
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left: 0px;
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}
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.divIntegrand {
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display:inline-block;
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position: relative;
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left: -8px;
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}
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.gridForPairOfIntegrationBounds {
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display:inline-grid;
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position: relative;
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top: -17px;
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left: -6px;
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grid-gap: 6px;
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text-align: left;
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}
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.lowerIntegrationBoundOfPair {
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grid-column: 1; grid-row: 2;
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}
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.upperIntegrationBoundOfPair {
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grid-column: 1; grid-row: 1;
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padding-left: 10px;
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}
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</style>",
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TeX=>""
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));
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# ===============================================================
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# Display the answer blanks properly in different modes
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Context()->texStrings;
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if ($displayMode eq 'TeX') {
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$integral1 = join("", (
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'\( \displaystyle \text{Volume = } \int_{ ',
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$xlow->ans_rule(5),
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' }^{ ',
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$xhigh->ans_rule(5),
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'}\)', $xhigh->ans_rule(30), '\(\, dx \)' ));
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} else {
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$integral1 = join("", (
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openDiv( { "class" => "divOnLineWithIntegrationLimits" } ),
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'\( \displaystyle \quad\quad \text{Volume = } \int \)',
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closeDiv(),
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openDiv( { "class" => "gridForPairOfIntegrationBounds" } ),
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openDiv( { "class" => "lowerIntegrationBoundOfPair" } ),
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$xlow->ans_rule(8),
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closeDiv(),
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openDiv( { "class" => "upperIntegrationBoundOfPair" } ),
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$xhigh->ans_rule(8),
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closeDiv(),
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closeDiv(),
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openDiv( { "class" => "divIntegrand" } ),
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$integrand->ans_rule(15),
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'\( \, dx \)',
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closeDiv(),
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) );
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}
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Context()->normalStrings;
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BEGIN_PGML
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>>[@ image(insertGraph($gr),width=>300, height=>300, tex_size=>400) @]*<<
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>>[`\quad`]Graph of [`y = [$a] - x^2`]<<
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a) A region of the Cartesian plane is shaded, as shown in the figure above. Use the Disk/Washer Method to set up an integral that gives the volume of the solid of revolution formed by revolving the region around the [`x`]-axis.
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[`\;`]
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[@ openDiv() . $integral1 . closeDiv() @]*
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b) Compute the volume of the solid. [_______________]
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END_PGML
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WEIGHTED_ANS( $xlow->cmp,10 );
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WEIGHTED_ANS( $xhigh->cmp ,10);
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WEIGHTED_ANS( $integrand->cmp->withPostFilter(AnswerHints(
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Formula("$f") => "This looks like you are finding the area of the region, not the volume of the solid of revolution.",
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[Formula("pi*$f"),Formula("($f)^2")] => "This is almost correct. What is the formula for the area of a circle?",
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)),50 );
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WEIGHTED_ANS( $rans->cmp,30 );
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Context("Fraction")->flags->set(reduceConstantFunctions => 0,);
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$a2 = $a*$a;
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$a3 = Fraction(2*$a,3);
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$c1 = Fraction(1,5);
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$c2 = Fraction(8,15);
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$c3 = Fraction(16,15);
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if ($a==4 or $a==9 or $a==16) {
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$w1 = $c2*sqrt($a)**5;
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$w2 = $c3*sqrt($a)**5;
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$c2 = "";
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$c3 = "";
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} else {
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$w1 = "($a)^{5/2}";
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$w2 = "($a)^{5/2}";
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}
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BEGIN_PGML_SOLUTION
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The curve [`y = [$f]`] intersects the [`x`]-axis where [`[$f]=0`], so at [`x = [$xlow]`] and [`x = [$xhigh]`]. A slice of the region rotated around the [`x`]-axis forms a disk of radius [`R(x) = [$f]`]. The cross-sectional area of the slice is [`\pi R(x)^2 = \pi([$f])^2`]. The volume is therefore given by the integral
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[``\int_{[$xlow]}^{[$xhigh]}\pi R(x)^2\;dx = \int_{[$xlow]}^{[$xhigh]}\pi([$f])^2\;dx \approx [$ans]``]
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Here is the calculation with the antidifferentiation carried out.
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[``\begin{aligned}
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\int_{[$xlow]}^{[$xhigh]}\pi([$f])^2\;dx &= \pi \int_{[$xlow]}^{[$xhigh]} \left([$a2]-[$a*2]x^2 + x^4\right)\;dx \\ \\
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&= \pi \left([$a2]x - [$a3]x^3 + [$c1]x^5\right)\bigg|_{[$xlow]}^{[$xhigh]} \\ \\
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&= \pi \left([$c2][$w1] - (-[$c2][$w1]\right) = [$c3][$w2]\pi
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\end{aligned}
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``]
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END_PGML_SOLUTION
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COMMENT('Randomization provides 13 different possible versions of this question.');
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ENDDOCUMENT();

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