Skip to content

Update two UMich double integral problems #1445

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
dlglin wants to merge 1 commit into
base: main
Choose a base branch
from
Open

Update two UMich double integral problems #1445

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
258 changes: 69 additions & 189 deletions OpenProblemLibrary/Michigan/Chap16Sec2/Q23.pg
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -2,10 +2,9 @@
# Problem from Calculus, multi-variable, Hughes-Hallett et al.,
# originally from 5ed (with updates)
# WeBWorK problem written by Gavin LaRose, <glarose@umich.edu>
# Updated for WW 2.21 by Danny Glin
# ENDDESCRIPTION

## Tagged by glr 03/25/10

## DBsubject(Calculus - multivariable)
## DBchapter(Integration of multivariable functions)
## DBsection(Double integrals over general regions)
Expand All @@ -32,207 +31,88 @@

DOCUMENT();

loadMacros(
"PGstandard.pl",
"PGchoicemacros.pl",
"MathObjects.pl",
"PGgraphmacros.pl",
"parserPopUp.pl",
"PGcourse.pl"
);
loadMacros("PGstandard.pl", "PGML.pl", 'parserGraphTool.pl', "PGcourse.pl");

Context("Numeric");
Context()->variables->add( y => 'Real' );
$showPartialCorrectAnswers = 1;
Context()->variables->add(y => 'Real');

## we plot a section of a circle with radius $r0
$r0 = random(2,5,1);
$dr = ( $r0 == 2 ) ? 1 : list_random(1,2);
## for an x- or y-range that extends betwee 0 and +/- $a
$a = $r0 - $dr;

## the x ranges for the different graphs are
@x0 = ( -1*$a, -1*$r0, 0, 0, -1*$r0, -1*$a );
@x1 = ( 0, 0, $a, $r0, 0, 0 );
## and the circle segments are plotted for
@xpl = ( "<-$a,0>", "<-$r0,-" . sqrt($r0*$r0-$a*$a) . ">", "<0,$a>",
"<" . sqrt($r0*$r0-$a*$a). ",$r0>",
"<-$r0,-" . sqrt($r0*$r0-$a*$a) . ">", "<-$a,0>" );
## and y ranges
@y0 = ( -1*($r0+1), -1*$r0, -1*($r0+1), -1*$r0, -1 -1 );
@y1 = ( 1, 1, 1, 1, $r0, ($r0+1) );
## x-ticks at
@xticks = ( -1*$a, -1*$r0, $a, $r0, -1*$r0, -1*$a );
## and y-ticks
@yticks = ( -1*$r0, -1*$a, -1*$r0, -1*$a, $a, $r0 );

## the extra lines we add connect the points
@pts = ( [ [0,-1*$r0], [0,0], [-1*$a,0], [-1*$a,-1*sqrt($r0*$r0 - $a*$a)] ],
[ [-1*$r0,0], [0,0], [0,-1*$a], [-1*sqrt($r0*$r0-$a*$a), -1*$a] ],
[ [0,-1*$r0], [0,0], [$a,0], [$a,-1*sqrt($r0*$r0 - $a*$a)] ],
[ [$r0,0], [0,0], [0,-1*$a], [sqrt($r0*$r0-$a*$a), -1*$a] ],
[ [-1*$r0,0], [0,0], [0,$a], [-1*sqrt($r0*$r0-$a*$a), $a] ],
[ [0,$r0], [0,0], [-1*$a,0], [-1*$a,sqrt($r0*$r0-$a*$a)] ] );
## and a point in the interior of the region to fill
@fillPt = ( [-.1,-.1], [-.1,-.1], [.1,-.1], [.1,-.1], [-.1,.1], [-.1,.1] );

## descriptions of the graphs
@desc = ( "Graph of part of the lower half of a circle of radius $r0, " .
"between x=-$a and x=0.",
"Graph of part of the left half of a circle of radius $r0, " .
"between y=-$a and y=0.",
"Graph of part of the lower half of a circle of radius $r0, " .
"between x=0 and x=$a.",
"Graph of part of the right half of a circle of radius $r0, " .
"between y=-$a and y=0.",
"Graph of part of the left half of a circle of radius $r0, " .
"between y=0 and y=$a.",
"Graph of part of the top half of a circle of radius $r0, " .
"between x=-$a and x=0." );

## the possible graphs are
@gr = ();

## build the graphs
for ( my $i=0; $i<6; $i++ ) {
$gr[$i] = init_graph( ($x0[$i]-1), $y0[$i], ($x1[$i]+1), $y1[$i],
axes=>[0,0], size=>[200,200] );
$gr[$i]->lb('reset');
$gr[$i]->moveTo( $xticks[$i], -.1 );
$gr[$i]->lineTo( $xticks[$i], 0.1, 'black' );
$gr[$i]->moveTo( -.1, $yticks[$i] );
$gr[$i]->lineTo( 0.1, $yticks[$i], 'black' );
$gr[$i]->lb( new Label( $xticks[$i], 0, "$xticks[$i]", 'black',
'left', 'top' ) );
$gr[$i]->lb( new Label( 0, $yticks[$i], "$yticks[$i]", 'black',
'right', 'top' ) );

if ( $i < 4 ) { $f = "-1*sqrt($r0^2 - x^2)"; }
else { $f = "sqrt($r0^2 - x^2)"; }

add_functions( $gr[$i], "$f for x in $xpl[$i] using " .
"color:blue and weight:2" );

$gr[$i]->moveTo( $pts[$i]->[0]->[0], $pts[$i]->[0]->[1] );
for ( my $j=1; $j<@{$pts[$i]}; $j++ ) {
$gr[$i]->lineTo( $pts[$i]->[$j]->[0], $pts[$i]->[$j]->[1], 'blue', 2 );
}
$gr[$i]->new_color( 'ltblue', 214, 230, 244 );
$gr[$i]->fillRegion( [$fillPt[$i]->[0], $fillPt[$i]->[1], 'ltblue'] );
}
$r0 = random(2, 5, 1);
$dr = ($r0 == 2) ? 1 : list_random(1, 2);
## for an x- or y-range that extends between 0 and +/- $a
$a = $r0 - $dr;

## display the graphs in this order
@grOrder = shuffle(6);
## then the correct answer is
for ( my $i=0; $i<6; $i++ ) {
if ( $grOrder[$i] == 0 ) {
$cor = ($i + 1);
last;
}
}
## and put them in a table
$grTab = begintable(3) .
row( "1." . image( insertGraph($gr[$grOrder[0]]), tex_size=>200,
height=>200, width=>200, extra_html_tags=>'alt="' .
$desc[$grOrder[0]] . '"' ),
"2." . image( insertGraph($gr[$grOrder[1]]), tex_size=>200,
height=>200, width=>200, extra_html_tags=>'alt="' .
$desc[$grOrder[1]] . '"' ),
"3." . image( insertGraph($gr[$grOrder[2]]), tex_size=>200,
height=>200, width=>200, extra_html_tags=>'alt="' .
$desc[$grOrder[2]] . '"' ) ) .
row( "4." . image( insertGraph($gr[$grOrder[3]]), tex_size=>200,
height=>200, width=>200, extra_html_tags=>'alt="' .
$desc[$grOrder[3]] . '"' ),
"5." . image( insertGraph($gr[$grOrder[4]]), tex_size=>200,
height=>200, width=>200, extra_html_tags=>'alt="' .
$desc[$grOrder[4]] . '"' ),
"6." . image( insertGraph($gr[$grOrder[5]]), tex_size=>200,
height=>200, width=>200, extra_html_tags=>'alt="' .
$desc[$grOrder[5]] . '"' ) ) .
endtable();

## the correct graph pop up
$grSelect = PopUp( [ '?',1,2,3,4,5,6], $cor );

$rsq = $r0*$r0;
$integrand = Formula("$a*x*y")->reduce();

$rsq = $r0 * $r0;

## the integral value
$intVal = Compute( "(1/4)*$a^3*$rsq - (1/8)*$a^5" );

Context()->texStrings;
TEXT(beginproblem());
BEGIN_TEXT

For the integral
\[
\int_{-$a}^{0} \int_{-\sqrt{$rsq-x^2}}^{0} $a x y\,dy\,dx,
\]
sketch the region of integration and evaluate the integral.
$PAR
Your sketch should be approximately the same as one of the
graphs shown below; which is the correct region?
Graph \{ $grSelect->menu() \}
$PAR
Then
\( \int_{-$a}^{0} \int_{-\sqrt{$rsq-x^2}}^{0} $a x y\,dy\,dx = \)
\{ ans_rule(35) \}

$PAR
Graphs:
$BR
$BCENTER
$grTab
$ECENTER

END_TEXT
Context()->normalStrings;

ANS($grSelect->cmp() );
ANS($intVal->cmp() );

$acu = $a*$a*$a;
$afi = $acu*$a*$a;
if ( $a == 2 ) {
$ao2 = '';
$intVal = Compute("(1/4)*$a^3*$rsq - (1/8)*$a^5");

$gt = GraphTool(
"{line,solid,(0,0),(0,-1)},
{line,solid,(0,0),(-1,0)},
{line,solid,(-$a,0),(-$a,-1)},
{circle,solid,(0,0),(0,-$r0)},
{fill,(-2/3,-1)}"
)->with(
availableTools => [
"LineTool", "CircleTool",
"VerticalParabolaTool", "HorizontalParabolaTool",
"FillTool"
],
snapSizeX => 1 / 2,
snapSizeY => 1 / 2
);

BEGIN_PGML
Consider the integral
[```\int_{-[$a]}^{0} \int_{-\sqrt{[$rsq]-x^2}}^{0} [$integrand]\,dy\,dx.```]

a. Sketch the region of integration
[_]{$gt}
b. Evaluate the integral.
[``\int_{-[$a]}^{0} \int_{-\sqrt{[$rsq]-x^2}}^{0} [$integrand]\,dy\,dx = ``][_]{$intVal}{35}
END_PGML

$acu = $a * $a * $a;
$afi = $acu * $a * $a;
if ($a == 2) {
$ao2 = '';
$acuo4 = 2;
$afio8 = 4;
} elsif ( $a/2 == int($a/2) ) {
$ao2 = $a/2;
$acuo4 = $acu/4;
$afio8 = $afi/8;
} elsif ($a / 2 == int($a / 2)) {
$ao2 = $a / 2;
$acuo4 = $acu / 4;
$afio8 = $afi / 8;
} else {
$ao2 = "\frac{$a}{2}";
$ao2 = "\frac{$a}{2}";
$acuo4 = "\frac{$acu}{4}";
$afio8 = "\frac{$afi}{8}";
}
($vn, $vd) = reduce( ($a*$a*$a*$rsq*2 - $afi), 8 );
$v = ( $vd == 1 ) ? "$vn" : "\frac{$vn}{$vd}";
($vn, $vd) = reduce(($a * $a * $a * $rsq * 2 - $afi), 8);
$v = ($vd == 1) ? "$vn" : "\frac{$vn}{$vd}";

Context()->texStrings;
SOLUTION(EV3(<<'END_SOLUTION'));
$PAR SOLUTION $PAR
BEGIN_PGML_SOLUTION
The region of integration is bounded by [`x = -[$a]`] and [`x = 0`],
below by [`y = -\sqrt{[$rsq] - x^2}`] and above by [`y = 0`]. The
correct region is therefore

[@ $gt->generateAnswerGraph(ariaDescription => "A circle of radius $r0 centered at (0,0), with the section to the right of the line x=-$a, below the x-axis and to the left of the y-axis shaded") @]*

The region of integration is bounded by \(x = -$a\) and \(x = 0\),
below by \(y = -\sqrt{$rsq - x^2}\) and above by \(y = 0\). The
correct region is therefore \{ $grSelect->correct_ans() \}.
Evaluating the integral, we have
\[
\int_{-$a}^{0}\int_{-\sqrt{$rsq-x^{2}}}^{0} $a x y\, dy dx =
\int_{-$a}^{0} $ao2\, x \, y^{2}\bigg|_{-\sqrt{$rsq-x^{2}}}^{0} dx
= \int_{-$a}^{0} -$ao2 x ($rsq - x^2)\,dx
\]
\[
-$ao2\int_{-$a}^{0} $rsq x - x^{3}\, dx
= -$ao2\left( $rsq(\frac12)(x^2) - (\frac14)(x^4)\right)\bigg|_{-$a}^{0}
\]
\[
= $acuo4\,$rsq - $afio8 = $v.
\]

END_SOLUTION
Context()->normalStrings;


;
[```
\int_{-[$a]}^{0}\int_{-\sqrt{[$rsq]-x^{2}}}^{0} [$integrand]\, dy dx =
\int_{-[$a]}^{0} [$ao2]\, x \, y^{2}\bigg|_{-\sqrt{[$rsq]-x^{2}}}^{0} dx
= \int_{-[$a]}^{0} -[$ao2] x ([$rsq] - x^2)\,dx
```]
[```
-[$ao2]\int_{-[$a]}^{0} [$rsq] x - x^{3}\, dx
= -[$ao2]\left( [$rsq](\frac12)(x^2) - (\frac14)(x^4)\right)\bigg|_{-[$a]}^{0}
```]
[```
= [$acuo4]\left([$rsq]\right) - [$afio8] = [$v].
```]
END_PGML_SOLUTION

ENDDOCUMENT();
71 changes: 25 additions & 46 deletions OpenProblemLibrary/Michigan/Chap16Sec7/Q25.pg
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -6,7 +6,6 @@

## Tagged by glr 05/28/10


## DBsubject(Calculus - multivariable)
## DBchapter(Integration of multivariable functions)
## DBsection(Change of variable)
Expand All @@ -31,70 +30,50 @@

DOCUMENT();

loadMacros(
"PGstandard.pl",
"PGchoicemacros.pl",
"MathObjects.pl",
"PGcourse.pl"
);

Context("Numeric");
$showPartialCorrectAnswers = 1;

$a = random(1,4,1);
$del = random(1,5,1);
$b = $a + $del;
loadMacros('PGstandard.pl', 'PGML.pl', 'PGcourse.pl');

$val = Compute( "$del*$del" );
Context('Numeric');

Context()->texStrings;
TEXT(beginproblem());
BEGIN_TEXT
$a = random(1, 4, 1);
$del = random(1, 5, 1);
$b = $a + $del;

Use the change of variables \( s=xy \), \( t=xy^2 \) to compute
\( \int_R xy^2\,dA \), where \( R \) is the region bounded by
\( xy=$a,\ xy=$b,\ xy^2=$a,\ xy^2=$b \).
$val = Compute($del**2);

$PAR
\( \int_R xy^2\,dA = \) \{ ans_rule(35) \}
BEGIN_PGML
Use the change of variables [` s=xy `], [` t=xy^2 `] to compute
[`` \iint_R xy^2\,dA ``], where [` R `] is the region bounded by
[` xy=[$a],\ xy=[$b],\ xy^2=[$a],\ xy^2=[$b] `].

END_TEXT
Context()->normalStrings;

ANS($val->cmp() );

Context()->texStrings;
SOLUTION(EV3(<<'END_SOLUTION'));
$PAR SOLUTION $PAR
[`` \iint_R xy^2\,dA = ``][_]{$val}{5}
END_PGML

BEGIN_PGML_SOLUTION
Given
\[
\begin{array}{ccc}s&=&xy\\t&=&xy^2,\end{array}.\]
[```
\begin{array}{ccc}s&=&xy\\t&=&xy^2,\end{array}.```]
we have
\[\frac{\partial(s,t)}{\partial(x,y)}=
[```\frac{\partial(s,t)}{\partial(x,y)}=
\left|
\begin{array}{ccc}
\frac{\partial s}{\partial x} &\frac{\partial s}{\partial y}\\
\frac{\partial t}{\partial x} &\frac{\partial t}{\partial y}
\end{array}\right|
=\left|\begin{array}{ccc}y&x\\y^2&2xy\end{array}\right|=xy^2=t.
\]
Since \[\frac{\partial(s,t)}{\partial(x,y)}\cdot
```]
Since [```\frac{\partial(s,t)}{\partial(x,y)}\cdot
\frac{\partial(x,y)}{\partial(s,t)}=1,
\qquad
\frac{\partial(x,y)}{\partial(s,t)}=\frac{1}{t}.
\]
```]

So
\[
\int_R xy^2\,dA=\int_{T}t\left|\frac{\partial(x,y)}{\partial(s,t)}\right|\,ds\,dt=
\int_{T}t\,(\frac 1t)\,ds\,dt=\int_{T}\,ds\,dt,\]
where \(T\) is the region bounded by \(s=$a\), \(s=$b\), \(t=$a\), \(t=$b\).
Then \[\int_Rxy^2\,dA=\int_$a^{$b}\,ds\int_$a^{$b}\,dt=$val.\]

END_SOLUTION
Context()->normalStrings;
[```
\iint_R xy^2\,dA=\iint_{T}t\left|\frac{\partial(x,y)}{\partial(s,t)}\right|\,ds\,dt=
\iint_{T}t\,(\frac 1t)\,ds\,dt=\iint_{T}\,ds\,dt,```]
where [`T`] is the region bounded by [`s=[$a]`], [`s=[$b]`], [`t=[$a]`], [`t=[$b]`].
Then [```\iint_Rxy^2\,dA=\int_{[$a]}^{[$b]}\,ds\int_{[$a]}^{[$b]}\,dt=[$val].```]

END_PGML_SOLUTION

;
ENDDOCUMENT();