** Using WeBWorK **

# DESCRIPTION
## Calculus
## ENDDESCRIPTION

## KEYWORDS ('integration','graph','area')
## Tagged by cmd6a 4/26/06

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Areas Between Curves')
## Date('')
## Author('')
## Institution('Rochester')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')
## TitleText2('Calculus: Early Transcendentals')
## EditionText2('1')
## AuthorText2('Rogawski')
## Section2('6.1')
## Problem2('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(3,10,1);
$b = random(2,8,1);
$c = random(-8,-2,1);
$d = random(.1,.95,.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');

$boundx1a = $boundx1 -1;
$boundx2a = $boundx2 +1;

#We put the functions here to keep plot_functions' entries simple
$fun1 = FEQ("$d*x^2 ? {$a} for x in [$boundx1a,$boundx2a] using color:red and weight:2");
$fun2 = FEQ("x for x in [$boundx1a,$boundx2a] using color:blue and weight:2");

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

($fun1_ref,$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 = -4 ;
$y1 = $d*$c**2 ;
$y2 = $d*$b**2 ;
$label_fun1 = new Label ($labelx,&{$fun1_ref->rule}($labelx),"f(x)",'red','left','bottom');
$label_fun2 = new Label ($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-.1,(&{$fun1_ref->rule}($c) + &{$fun2_ref->rule}($c))/2,"x=$c",'black','right','middle');
$label_line2 = new Label ($b+.1,(&{$fun1_ref->rule}($b) + &{$fun2_ref->rule}($b))/2,"x=$b",'black','left','middle');

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

TEXT(image( insertGraph($graph), 'width' => 200, 'height' => 200) );

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