##DESCRIPTION
##KEYWORDS('integrals', 'improper')
## sbrummel tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('13,14')
##ENDDESCRIPTION

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(2,9,1);
$b = -1+($a);
$test = (-1+((-1)**$a))/2;

if ($test==-1) {
        $soln = 'INF';
} else {
        $soln = $test;
}

TEXT(EV2(<<EOT));

Determine whether the integral is divergent or convergent.
If it is convergent, evaluate it.
If it diverges to infinity, state your answer as "INF" (without the
quotation marks).
If it diverges to negative infinity, state your answer as "MINF".
If it diverges without being infinity or negative infinity, state your
answer as "DIV".

$BR \[ \int_{-\infty}^{\infty} x^$b  e^{-x^$a} dx \]
$BR \{ans_rule( 30) \}
$BR
EOT
ANS(num_cmp($soln, strings=>['INF', 'MINF', 'DIV']));

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

##DESCRIPTION
##KEYWORDS('integrals', 'improper')
## sbrummel tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('24')
##ENDDESCRIPTION

## 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(1,6,1);
$soln = arctan($a) + arcsin(1);

TEXT(EV2(<<EOT));
Determine whether the integral is divergent or convergent.
If it is convergent, evaluate it. If not, state your answer as "divergent."
$BR \[ \int_{-\infty}^{$a}  \frac{1}{x^2 + 1} dx \]
$BR \{ans_rule( 30) \}
$BR
EOT
ANS(num_cmp($soln));

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

##DESCRIPTION
##KEYWORDS('integrals', 'improper')
## sbrummel tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('21')
##ENDDESCRIPTION

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(2,7,1);
$soln = "divergent";

TEXT(EV2(<<EOT));
Determine whether the integral is divergent or convergent.
If it is convergent, evaluate it. If not, state your answer as "divergent."
$BR \[ \int_{$a}^{\infty} \frac{\ln(x)}{x} dx \]
$BR \{ans_rule( 30) \}
EOT
ANS(str_cmp($soln));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('integrals', 'improper')
# sbrummel tagged
## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('3,4')
##ENDDESCRIPTION

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(5, 9);
$b=random(-2,-.5,.5);
$d=$b-1;
$n=-$b;
$soln3 = ($a)**($b);
$u=10**$b;
$soln1 = $soln3-$u;
$soln2 = $soln3-($u*$u);

$showPartialCorrectAnswers = 1;

TEXT(EV2(<<EOT));

Find the area under the curve
\[ y = $n x^{$d} \]
$BR
from \( x = $a\ \) to \(\ x = t \) and evaluate it for \(t = 10\ \), \(\ t = 100\).
$BR
Then find the total area under this curve for \( x \geq $a \).
$BR
(a) t = 10
$BR \{ans_rule( 30) \}
$BR
EOT
ANS(num_cmp($soln1));

TEXT(EV2(<<EOT));
$BR
(b) t = 100
$BR \{ans_rule(30)\}
$BR
EOT
ANS(num_cmp($soln2));

TEXT(EV2(<<EOT));
$BR
(c) Total area
$BR \{ans_rule(30)\}
$BR
EOT
ANS(num_cmp($soln3));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('integrals', 'improper')
## sbrummel tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('8')
##ENDDESCRIPTION

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(2,9,1);
$soln = $a * 2 / (5**(.5));

TEXT(EV2(<<EOT));

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent."

$BR \[ \int_{2}^{\infty}  \frac{$a}{(x+3)^{3/2}} dx \]
$BR $BR \{ans_rule( 30) \}
$BR
EOT
ANS(num_cmp($soln));

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

##DESCRIPTION
##KEYWORDS('integrals', 'improper')
## sbrummel tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('1,2')
##Ellis & Gulick section 7.7
##Authored by Zig Fiedorowicz 5/17/2000
##ENDDESCRIPTION

DOCUMENT();

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

install_problem_grader(~~&std_problem_grader);

$showPartialCorrectAnswers = 0;

$aa = random(4,10,1);
$a2 =  2*$aa+1;
$bb = random(3, 8,1);
$b2 = $bb*$bb;
$cc = random(5,12,1);
$dd = random(7,45,1);

TEXT(beginproblem());
BEGIN_TEXT
Consider the following integrals. Label each as ``P'', ``C'', ``D'', according
as the integral is proper, improper but convergent, or improper and divergent.
$BR
END_TEXT

$questStr1 = EV2("\( \displaystyle \int_{-\infty}^{\infty}\sin($bb x)\,dx\)");
$ansStr1 = "D";
$questStr2 = EV2("\( \displaystyle \int_{-\pi/$aa}^{$a2\pi/2}\tan^2($bb x)\,dx\)");
$ansStr2 = "D";
$questStr3 = EV2("\( \displaystyle \int_{-\infty}^{\infty}\frac{x}{x^2+$cc}\,dx\)");
$ansStr3 = "D";
$questStr4 = EV2("\( \displaystyle \int_{$bb}^{\infty}\frac{1}{\sqrt{t^2-$b2}}\,dt\)");
$ansStr4 = "D";
$questStr5 = EV2("\( \displaystyle \int_{0}^{$a2}\frac{1}{\sqrt[3]{x-$aa}}\,dx\)");
$ansStr5 = "C";
$questStr6 = EV2("\( \displaystyle \int_{-$cc\pi}^{$dd\pi}\sin(x)\tan^{-1}(x)\,dx\)");
$ansStr6 = "P";
if ($cc >8) {
   $questStr7 = EV2("\( \displaystyle \int_{1}^{\infty}s e^{-$bb s^2}\,ds\)");
   $ansStr7 = "C";
} else {
   $questStr7 = EV2("\( \displaystyle \int_{1}^{\infty}s e^{$bb s^2}\,ds\)");
   $ansStr7 = "D";
}
$questStr8 = EV2("\( \displaystyle \int_{$aa}^{$a2}\ln(x-$aa)\,dx\)");
$ansStr8 = "C";

@questions =( $questStr1,$questStr2,$questStr3,$questStr4,$questStr5,$questStr6,
$questStr7,$questStr8);

@answers =( $ansStr1,$ansStr2,$ansStr3,$ansStr4,$ansStr5,$ansStr6,
$ansStr7,$ansStr8);

@shuffle = &shuffle(scalar(@questions));

TEXT(
&match_questions_list(@questions[@shuffle])
);

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
$PAR
This problem covers section 7.7 of the text. For further practice,
you should do 5-10 odd numbered problems from among problems 1-59 on
pages 486-487.
END_TEXT
}
}

ANS(str_cmp([@answers[@shuffle] ]));


ENDDOCUMENT();
________________________________________________________________________________

##KEYWORDS('integrals', 'improper')
##DESCRIPTION
## Determine if an improper integral exists and evaluate it.
##ENDDESCRIPTION

## Shotwell cleaned


## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('30')

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

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

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

$a = random(8,13,1);
$b = random(1, 7);

$soln = "3/2*$a*($a-$b)^(2/3)";

BEGIN_TEXT
Determine whether the integral is divergent or convergent.
If it is convergent, evaluate it. $BR$BR
If it diverges to infinity, state your answer as $BITALIC inf $EITALIC . 
If it diverges to negative infinity, state your answer as $BITALIC -inf $EITALIC . If it diverges without being infinity or negative infinity, state your answer as $BITALIC div $EITALIC . 

$BR \[ \int_{$b}^{$a} \frac{$a}{\sqrt[3]{x-$b}} dx \]
$BR Answer: \{ans_rule( 30) \}
$BR
END_TEXT

ANS(num_cmp($soln, strings=>['inf', '-inf', 'div']));

ENDDOCUMENT();
________________________________________________________________________________

##KEYWORDS('integrals', 'improper')
##DESCRIPTION
## Determine if an improper integral converges and evaluate it.
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('33')

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

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

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

$a = -random(1,20,1);
$b = random(1,20,1);
$c = random(2,8,2);

BEGIN_TEXT

Determine whether the integral is divergent or convergent.
If it is convergent, evaluate it. $BR$BR
If it diverges to infinity, state your answer as $BITALIC inf $EITALIC .
If it diverges to negative infinity, state your answer as $BITALIC -inf $EITALIC . If it diverges without being infinity or negative infinity, state your answer as $BITALIC div $EITALIC .

$BR \[ \int_{$a}^{$b}  \frac{1}{x^{$c}} \, dx \]
$BR Answer: \{ans_rule(40) \}
$BR
END_TEXT

ANS(num_cmp('inf', strings=>['inf', '-inf', 'div']));


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

##DESCRIPTION
##KEYWORDS('integrals', 'improper')
## sbrummel tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('20')
##ENDDESCRIPTION

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;

$l = random(.1,.9,.1);
$m = random(-1,1,2);
$n = 1+($l*$m);
$a=random(2,8,1);
$b=random(2,5,1);
$soln = (1+$a*$b)*exp(-$a*$b)/$b**2;

TEXT(EV2(<<EOT));

Determine whether the integral is divergent or convergent.
If it is convergent, evaluate it. If not, give the answer -1.
$BR \[ \int_{$a}^{\infty} x e^{-${b}x} dx \]
$BR \{ans_rule( 30) \}
EOT
ANS(num_cmp($soln));

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

##DESCRIPTION
##KEYWORDS('integrals', 'improper')
## sbrummel tagged and PAID on 2-20-2004
## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Improper Integrals')
## Date('6/3/2002')
## Author('Arnie Pizer')
## Institution('rochester')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.8')
## Problem1('27')
##ENDDESCRIPTION

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;

$l = random(.1,.9,.1);
$m = random(-1,1,2);
$n = 1+($l*$m);
$a=random(2,8,1);
$test = ((-1-$m)/2)+((1-$m)/2)*($l**-1)*($a**($l));

if ($test==-1) {
	$soln = 'INF';
} else {
	$soln = $test;
}

TEXT(EV2(<<EOT));

Determine whether the integral is divergent or convergent.
If it is convergent, evaluate it.
If it diverges to infinity, state your answer as "INF" (without the
quotation marks).
If it diverges to negative infinity, state your answer as "MINF".
If it diverges without being infinity or negative infinity, state your
answer as "DIV".

$BR \[ \int_{0}^{$a} \frac{1}{x^{$n}} dx \]
$BR \{ans_rule( 30) \}
EOT
ANS(num_cmp($soln, strings=>['INF', 'MINF', 'DIV']));

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