#KEYWORDS('Integral', 'Volume')
##DESCRIPTION
##  Compute the volume of a rotation
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('6.2')
## Problem1('2,3')



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

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

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

$a = random(.1,1,.1);
$b = random(1,5,1);
$c = random(1,5,1);
$answer = "pi*((2*$b)^(-1)*(e^(2*$b*$a)-1)+2*($c/$b)*(e^($a*$b)-1)+($c)^2*$a)";

BEGIN_TEXT

Find the volume of the solid formed by rotating the region enclosed by $BR
\[y=e^{$b x} + $c, \ \ y=0, \ \ x=0, \ \ x=$a\] 
about the \(x\)-axis. $BR$BR
Answer:\{ans_rule(60)\}

END_TEXT

ANS(num_cmp($answer));

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

## DESCRIPTION
## Calculus: Volumes
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'volumes')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.2')
## Problem1('7')

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;

$c = random(2,5,1)  ;
$b = random(2,4,1)  ;
$a = ($c)**($b-1);

TEXT(EV2(<<EOT));
Find the volume of the solid formed by rotating the region inside the          first quadrant enclosed by $BR
\[ y = x^$b , \quad y = $a x \] $BR
about the \(x\)-axis. $BR
Volume = \{ans_rule( 25) \}
EOT

$answer=4*arctan(1)*((1/3)*($a)**2*($c)**3-(2*$b+1)**(-1)*($c)**(2*$b+1));
ANS(num_cmp($answer));

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

## DESCRIPTION
## Calculus: Volumes
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'volumes')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.2')
## Problem1('14')

## Before doing anything, we must import the macro definitions on the next few lines.

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;

my $pi = arccos(-1);
$n = random(3,6,1);
$a = random(1,4,1);
$b = random(5,9,1);

$soln = 2*$pi*($b**(2-$n) - $a**(2-$n)) / (2-$n);

TEXT(EV2(<<EOT));

Find the volume of the solid obtained by rotating the region bounded by the
given curves about the specified axis.

\[ y = 1/x^{$n}, \quad y = 0, \quad x = $a, \quad x = $b
\] 
about the \(y\)-axis.


$BR 
Volume = \{ans_rule( 25) \}
$BR
EOT

ANS(num_cmp($soln));

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

## DESCRIPTION
## Calculus: Volumes
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'volumes')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.2')
## Problem1('1')


## Before doing anything, we must import the macro definitions on the next few lines.

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);
$pi = arccos(-1);
$soln =$a**2*$pi/5;

TEXT(EV2(<<EOT));

Find the volume of the solid obtained by rotating the region bounded by the
given curves about the specified axis.
 \[ y=$a x^2, \quad x = 1, \quad y = 0, \]
about the \(x\)-axis.

$BR
Volume = \{ans_rule( 25) \}
$BR
EOT
ANS(num_cmp($soln));

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

## DESCRIPTION
## Calculus: Volumes
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'volumes')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.2')
## Problem1('2')

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(.1,1,.1);
$b = random(2,5,1);
$c = random(1,5,1);
$pi = 4*arctan(1);

TEXT(EV2(<<EOT));
Find the volume of the solid formed by rotating the region enclosed by $BR
\[y=e^{$b x} + $c, \quad y=0, \quad x=0, \quad x=$a \] $BR
about the \(x\)-axis.
$BR
Volume = \{ans_rule( 25) \}
EOT

$answer = $pi*((2*$b)**(-1)*(exp(2*$b*$a)-1)+2*($c/$b)*(exp($a*$b)-1)+($c)**2*$a);
ANS(num_cmp($answer));

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

## DESCRIPTION
## Calculus: Volumes
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'volumes')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('6.2')
## Problem1('47')

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;

$h = random(1, 50,1) ;
$r = random($h+1,100,1) ;


TEXT(EV2(<<EOT));
Find the volume of a cap of a sphere with radius \( r=$r\) and height \( h=$h\).

$BR

Volume=\{ans_rule(50)\}
EOT

$answer = $PI*$h**2*($r-$h/3);
ANS(num_cmp($answer));

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

##DESCRIPTION
##KEYWORDS('integrals', 'volume')
## kshort tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('6.4')
## Problem1('54')
##Ellis and Gullick: section 8.1
##Authored by Zig Fiedorowicz 5/19/2000
##ENDDESCRIPTION

DOCUMENT();

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

##Note this uses Mike Gage's custom full_partial_grader
##contained in file PGgraders.pl
install_problem_grader(~~&full_partial_grader);

$showPartialCorrectAnswers = 1;

$aa = random(2,6);
$a2 = $aa*$aa;
$bb = random(2,6);
if ($bb==$aa) {$bb++;}
$b2 = $bb*$bb;

TEXT(beginproblem());

BEGIN_TEXT
$BR
    
\{image("osu_in_20_3.gif", width=>249, height=>122)\}
    
$BR
The base of a certain solid is the area bounded above by the graph of \(y=f(x)=$a2\)
and below by the graph of \(y=g(x)= $b2 x^2\). Cross-sections perpendicular to the \(x\)-axis
are squares. (See picture above, click for a better view.)
$BR
Use the formula
\[V=\int_a^b A(x)\,dx\]
to find the volume of the solid.
$BR
{\bf Note:} You can get full credit for this problem by just entering the final
answer (to the last question) correctly. The initial questions are meant as hints
towards the final answer and also allow you the opportunity to get partial credit.
$BR
The lower limit of integration is \(a\) =  \{ ans_rule()\}
$BR

The upper limit of integration is \(b\) =  \{ ans_rule()\}
$BR

The side \(s\) of the square cross-section is the following function of \(x\):
\{ ans_rule(40)\}
$BR

\(A(x)\)= \{ ans_rule(40)\}
$BR

Thus the volume of the solid is \(V\) =  \{ ans_rule()\}
END_TEXT

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
$PAR
This problem is similar to problems 29-34 of section 8.1 of the text.
END_TEXT
}
}

ANS(num_cmp(-$aa/$bb));
ANS(num_cmp($aa/$bb));
ANS(fun_cmp("$a2-$b2*x^2", vars=>"x"));
ANS(fun_cmp("($a2-$b2*x^2)^2", vars=>"x"));
ANS(num_cmp((16/15)*$aa**5/$bb));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('integrals', 'volume')
## kshort tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('6.4')
## Problem1('53')
##Ellis and Gullick: section 8.1
##Authored by Zig Fiedorowicz 5/19/2000
##ENDDESCRIPTION

DOCUMENT();

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

$showPartialCorrectAnswers = 1;

$bb = random(2,5);
$aa = $bb+random(1,3);
$aa = 10*$aa;
$a2 = $aa*$aa;
$bb = 10*$bb;
$b2 = $bb*$bb;

TEXT(beginproblem());
BEGIN_TEXT
As viewed from above, a swimming pool has the shape of the ellipse
\[\frac{x^2}{$a2}+\frac{y^2}{$b2}=1\]
The cross sections perpendicular to the ground and parallel to the \(y\)-axis
are squares. Find the total volume of the pool. (Assume the units of length and
area are feet and square feet respectively. Do not put units in your answer.)
$BR
\(V\) = \{ ans_rule()\}
END_TEXT

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
$PAR
This is similar to problem 47 in section 8.1 of the text.
END_TEXT
}
}

ANS(num_cmp((16/3)*$b2*$aa));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('integrals', 'volume')
## kshort tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Applications of Integration')
## DBsection('Volumes')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('6.4')
## Problem1('69')
##Ellis and Gullick: section 8.1
##Authored by Zig Fiedorowicz 5/19/2000
##ENDDESCRIPTION

DOCUMENT();

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

$showPartialCorrectAnswers = 1;
if (!($studentName =~ /PRACTICE/)) {
$seed = random(1,4,1);
if ($studentName =~ /VINCE VERSION1/) {$seed = 1;}
if ($studentName =~ /VINCE VERSION2/) {$seed = 2;}
if ($studentName =~ /VINCE VERSION3/) {$seed = 3;}
if ($studentName =~ /VINCE VERSION4/) {$seed = 4;}
SRAND($seed);}


$aa = random(5,10);
$bb = random(2,$aa-2);
$pi = 4*atan(1,1);

TEXT(beginproblem());
BEGIN_TEXT
A soda glass has the shape of the surface generated by revolving
the graph of \(y=$aa x^2\) for \(0\le x\le 1\) about the \(y\)-axis.
Soda is extracted from the glass through a straw at the rate of
\(1/2\) cubic inch per second. How fast is the soda level in the glass
dropping when the level is $bb inches? (Answer should be implicitly in units of
inches per second. Do not put units in your answer. Also your answer should be
positive, since we are asking for the rate at which the level DROPS rather than
rises.)
$BR
answer: \{ ans_rule()\}
END_TEXT

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
$PAR
This is similar to problem 52 in section 8.1 of the text.
END_TEXT
}
}

ANS(num_cmp($aa/(2*$pi*$bb)));

ENDDOCUMENT();