## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('6')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(2, 5);
$add=random(1, 5);
$b=$a+$add;
$e = exp(1);
$soln = ($e**$b)/$b-($e**$a)/$a-(1/$b)+(1/$a);

TEXT(EV2(<<EOT));
Sketch the region enclosed by the curves given below.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
\[ y=e^{$a x}, \quad  y=e^{$b x}, \quad  x=1 \]
Area =  \{ans_rule( 25) \}
EOT

ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'graph', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('1')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl",
"PGgraphmacros.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

#here are the parameters that will be varied.
$a = random(2,10,1);
$b = random(2,8,1);
$c = random(-8,-2,1);
$d = random(.15,.85,.1);

#with $bound* we ensure that the graph of the function will always be in view.
$boundx1 = $c-3;
$boundy1 = $c-8;
$boundx2 = $b +2;
$boundy2 = $d*8.2**2 +$a;

#here we create start and stop points for the x=lines which will be drawn later.
$line1_finish_y = $d*$c**2 +$a ;
$line2_finish_y = $d*$b**2 +$a ;

@line1_start = ($c, $c);
@line1_finish = ($c,$line1_finish_y,'black');
@line2_start = ($b,$b);
@line2_finish = ($b,$line2_finish_y,'black');

#We put the functions here to keep plot_functions' entries simple
$fun1 = FEQ("$d*x^2 ? {$a} for x in [$boundx1,$boundx2] using color:red and weight:2");
$fun2 = FEQ("x for x in [$boundx1,$boundx2] using color:red and weight:2");

$graph = init_graph($boundx1,$boundy1,$boundx2,$boundy2,'axes' => [0,0],
    'height' => 300,
        'width'  => 300
    );

($fun1_ref,$fun2_ref) = plot_functions ($graph,
    $fun1, $fun2
            );
#Here Professor Gage uses a new routine for drawing non-functions (parametrically).
#And replaces the first line. However, this doesnt work in the system version,
#since the PGgraphmacros.pl (?) hasn't been updated there yet).

$temp = new Fun (sub {$c;},
         sub {my $x = shift;
             $c+($x)*($line1_finish_y - $c);},
         $graph);
$temp ->color('black');
$temp -> domain(0,1);
$temp -> weight(2);
# here I make the two vertical lines at x=$c and x=$b
#line1 (x<0)
#$graph->moveTo(@line1_start);
#$graph->lineTo(@line1_finish);
#line2 (x>0)
$graph->moveTo(@line2_start);
$graph->lineTo(@line2_finish);

#where to put the labels
#$label_lineN = new Label ($x,$y,$string,$color,@justification)
#$string is the the text of the label, @justification can be: right, left or center.

$labelx = -5 ;
$y1 = $d*$c**2 ;
$y2 = $d*$b**2 ;
$label_fun1 = new Label ($labelx,&{$fun1_ref->rule}($labelx),"f(x)",'blue','left');
$label_fun2 = new Label ($labelx,&{$fun2_ref->rule}($labelx),"g(x)",'blue','left');
# NOTE: $fRef->rule is a reference to the subroutine which calculates the function.  It was
# defined in the output of plot_functions. It is used here to calculate the y value of the
# label corresponding to the function, and below to find the y values for the labels
# corresponding to the first and second derivatives.

$label_line1 = new Label ($c,$y1,"x=$c",'black','center');
$label_line2 = new Label ($b,$y2,"x=$b",'black','center');

#prints the labels
$graph->lb($label_fun1,$label_fun2 ,$label_line1 ,$label_line2 );

TEXT(image( insertGraph($graph), 'width' => 175, 'height' => 175) );

TEXT(EV2(<<EOT));
$PAR
Find the area enclosed between
 \( f(x) = $d x^2 ? {$a} \)
and
 \( g(x) = x \), on the interval
from \(x= $c  \)  to
\(x= $b\).
 $BR
Area = \{ans_rule(40)\}

EOT

$answer = ($d /3)*($b**3 -$c**3) + .5*($c**2 -$b**2) + $a*($b -$c);
ANS(num_cmp($answer));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('9')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(2, 7);
$soln = log($a)+1/$a-1;

TEXT(EV2(<<EOT));

Sketch the region enclosed by the given curves.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
\[ y=\frac{1}{ x}, \quad  y=\frac{1}{ x^2},\quad  x=$a \]
Area =  \{ans_rule( 25) \}
EOT

ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('10')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1.5,2.5,0.5);
$b = random(3,5,1);
$m = random(-2,-0.5,0.5);

$c1 = 2*$a;
$c2 = -2*$m;
$c3 = 2*($a - $m);

$soln = ($b**3 - $a**3)/(3*$a**2) - ($b**2 - $a**2)/(2*$m) + ($b - $a)*($a/$m - 1);

TEXT(EV2(<<EOT));
Sketch the region enclosed by the curves given below.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
 \[ 2 y = $c1 \sqrt{x} , \quad   y = $b,  \quad \mathrm{and} \quad 2 y + $c2 x = $c3 \]
$BR
Area = \{ans_rule( 25) \}
EOT

ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('13')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(1, 7);
$b=random(2, 7);
$x=sqrt($a/($b-1));
$soln = 2*((-$b+1)/3*$x**3+$a*$x);

TEXT(EV2(<<EOT));
Sketch the region enclosed by the given curves.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
\[ y=$b x^2,\quad  y=x^2 + $a \]
Area =  \{ans_rule( 25) \}
$BR
EOT
ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('17')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$b=random(2, 8);
$a= 1 - $b;
$c= - $a * $b;

$soln = $c * ($b - $a) + ($b**2 - $a**2)/2 - ($b**3 - $a**3)/3;


TEXT(EV2(<<EOT));

Sketch the region enclosed by the given curves.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
$BR \[ x + y^2 = $c, \quad  x + y = 0 \]
Area =  \{ans_rule( 25) \}
$BR
EOT
ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('19')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$b=random(1, 12);
$a= $b**2;

$soln = 4*($a*$b-($b**3)/3);


TEXT(EV2(<<EOT));

Sketch the region enclosed by the curves given below.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
\[ x= $a-y^2, \quad  x = y^2 - $a \]
$BR
Area = \{ans_rule( 60) \}
$BR
EOT
ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('20')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(2, 9);
$b=random(2, 9);
$soln = 2*($b**2-$a/sqrt(2));

TEXT(EV2(<<EOT));
Sketch the region enclosed by the given curves.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
$BR \[ y = $a \cos x, \quad y = ($b \sec x)^2, \quad  x = -\pi / 4, \quad  x = \pi / 4 \]
Area =  \{ans_rule( 25) \}
EOT
ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('21')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(2, 16,2);
$b=random($a/2+1,10,1);
$soln = $a**2/4/$b;

TEXT(EV2(<<EOT));
Sketch the region in the first quadrant enclosed by the given curves.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
$BR \[ y = $a \cos x, \quad  y = $b \sin 2 x, \quad   x = 0.  \]
Area =  \{ans_rule( 25) \}
EOT
ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: Areas between Curves
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integration', 'integrals', 'areas between curves')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.1')
## Problem1('23')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(2, 9);
$b=2*$a;
$soln = $a*(2-$PI/2);

TEXT(EV2(<<EOT));
Sketch the region enclosed by the curves given below.
Decide whether to integrate with respect to \(x\) or \(y\).
Then find the area of the region.
\[ y = $a \cos x, \quad y = $a-\frac{$b x}{ \pi}.   \]
$BR
Area = \{ans_rule(30) \}
EOT
ANS(num_cmp($soln));

ENDDOCUMENT();        # This should be the last executable line in the problem.