## DESCRIPTION
##  Calculus: The Definite Integral
## ENDDESCRIPTION

## KEYWORDS('calculus', 'definite integral')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('5/30/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('36')


DOCUMENT();

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

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

$a = random(2,6,1);
$b = random(2,9,1);
$d = random(4,10,1);

TEXT(EV2(<<EOT));

Evaluate the integral by interpreting it in terms of areas:
$BR
$PAR
\( \displaystyle{ \int_{0}^{$d}  |$b x - $a| \, dx} = \)
$PAR
   \{ans_rule(25) \}
$BR
$BR

EOT

$ans = $a**2/(2*$b) + ($d - $a/$b)*($b*$d - $a)/2;

ANS(num_cmp($ans));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('integrals', 'Riemann Sums', )
##  Approximate area under a curve using riemann sums
##ENDDESCRIPTION

## BenBush tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('1,2,3,4')

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(1,5,1);      ## left endpoint
$dx = random(1,2,1);        ## delta x
$n = random(3,4,1);
$b = $a + $n*$dx;            ## right endpoint, n subdivisions

$c = random(2,4,1);

sub myfunc {
    $x = shift @_;
    return $x**2 + $c*$x;
}

BEGIN_TEXT

Estimate the area under the graph of \( f(x)= x^2 + $c x \) from \( x=$a \) to \( x=$b \) 
using \( $n \) approximating rectangles and left endpoints. 

$BR \{ans_rule(30)\}

END_TEXT

if ($n == 3) { 
	$ans = $dx* (myfunc($a)+myfunc($a+$dx)+myfunc($a+2*$dx));
} 

if ($n == 4) {
	$ans = $dx* (myfunc($a)+myfunc($a+$dx)+myfunc($a+2*$dx)+myfunc($a+3*$dx));
}

ANS(num_cmp($ans));

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

##DESCRIPTION
##KEYWORDS('integrals', 'theory', 'Riemann sums')
##Recognising Riemann sums
##Authored by Zig Fiedorowicz 1/19/2000
##ENDDESCRIPTION

## BenBush tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('15,16,17,18,63,64')

DOCUMENT();

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


$showPartialCorrectAnswers = 1;

$a = random(2,7,1);
$b = $a + random(1,4,1);
$c = random(2,6,1);
$c2 = 2*$c;
$c3 = 3*$c;
$answ = 2*($b+$c)**1.5/3 - 2*$b**1.5/3;

TEXT(beginproblem());
BEGIN_TEXT
The following sum
\[
\sqrt{$b + \frac{$c}{n}} \cdot \left( \frac{$c}{n}\right) + 
\sqrt{$b + \frac{$c2}{n}} \cdot \left( \frac{$c}{n}\right) +
\ldots + \sqrt{$b + \frac{$c n}{n}} \cdot \left( \frac{$c}{n}\right)
\]

is a right Riemann sum for the definite integral
\[\int_{$a}^b f(x)\, dx\]
$BR

where \(b\) = \{ ans_rule()\}
$BR

and \(f(x)\) =  \{ ans_rule()\}
$PAR

It is also a Riemann sum for the definite integral
\[\int_{$b}^c g(x)\, dx\]
$BR

where \(c\) = \{ ans_rule()\}
$BR

and \(g(x)\) =  \{ ans_rule()\}
$PAR

The limit of these Riemann sums as \(n \to \infty\) is
\{ ans_rule(40)\}
END_TEXT

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
$PAR
{\bf Hint:} This is similar to Problem 7 in WeBWorK Problem Set Hmwk \#1.
END_TEXT
}
}

ANS(num_cmp($a + $c));
ANS(fun_cmp("sqrt($b - $a + x)", vars=>"x"));
ANS(num_cmp($b + $c));
ANS(fun_cmp("sqrt(x)", vars=>"x"));
ANS(num_cmp($answ));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('integrals', 'theory', 'Riemann sums')
##Ellis & Gulick section 5.2 - Recognising Riemann sums
##Final exam review
##Authored by Zig Fiedorowicz 3/7/2000
##ENDDESCRIPTION

DOCUMENT();

## BenBush tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('15,16,17,18,63,64')

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

$showPartialCorrectAnswers = 1;

$a = random(2,7,1);
$aa = $a*$a;
$a2 = 2*$a;
$a3 = 3*$a;
$answ = 3.141592654*$aa/4;

TEXT(beginproblem());
BEGIN_TEXT
The following sum $BR
\( 
\sqrt{$aa - \left(\frac{$a}{n}\right)^2} \cdot  \frac{$a}{n}  + 
\sqrt{$aa - \left(\frac{$a2}{n}\right)^2} \cdot  \frac{$a}{n}  +
\ldots + \sqrt{$aa - \left(\frac{$a n}{n}\right)^2} \cdot  \frac{$a}{n} 
\) $BR
is a right Riemann sum for the definite integral
\[\int_0^b f(x)\, dx\]
where \(b\) = \{ ans_rule()\}
$BR
and \(f(x)\) =  \{ ans_rule()\}
$PAR
The limit of these Riemann sums as \(n \to \infty\) is
\{ ans_rule()\}
END_TEXT

ANS(num_cmp($a, mode=>"strict", relTol=>0));
ANS(fun_cmp("sqrt($aa - x^2)", vars=>"x"));
ANS(num_cmp($answ));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##  Evaluate an integral by interpreting it in terms of areas
##ENDDESCRIPTION
##KEYWORDS('integrals', 'theory')

## Shotwell cleaned
## lcao , PAID on 11-24-2003

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('32')

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, 7, 1);

BEGIN_TEXT
Evaluate the integral below by interpreting it in terms of areas.
In other words, draw a picture of the region the integral
represents, and find the area using high school geometry.
\[ \int_{-$a}^{$a} \sqrt{\{$a**2\} - x^2 }dx \]
Answer: \{ans_rule( 30)\}
END_TEXT

$answer="($a^2)*pi/2";

ANS(num_cmp($answer));

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

##DESCRIPTION
##  Properties of the definite integrals
##ENDDESCRIPTION
##KEYWORDS('integrals', 'theory')

## Shotwell cleaned
## lcao , PAID on 11-24-2003

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('37')

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, 5, 1);
$b= random(6, 10, 1);
$c= non_zero_random(-70, 70, 1);

BEGIN_TEXT
If \( \displaystyle \int_{$a}^{$b} f(x) dx = $c \), then 
\(\displaystyle \int_{$b}^{$a} f(t) dt = \) \{ans_rule(10)\}
END_TEXT

ANS(num_cmp(-$c));

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

##DESCRIPTION
##  Evaluate a definite integral
##ENDDESCRIPTION
##KEYWORDS('integrals', 'theory')

## Shotwell cleaned
## lcao , PAID on 11-24-2003

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('38')

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, 5, 1);
$b= random(6, 10, 1);
$c= random(2, 70, 1);

BEGIN_TEXT
Evaluate \(\displaystyle \int_{$a}^{$a} ( x^{$a}\cos($b x) e^{$b x} + $c) dx .\)
$BR$BR
Answer: \{ans_rule(10)\}
END_TEXT

ANS(num_cmp(0));

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

##DESCRIPTION
##  Write a difference of two integrals as a single integral
##ENDDESCRIPTION
##KEYWORDS('integrals', 'theory')

## Shotwell cleaned
## lcao , PAID on 11-24-2003

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('44')

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, 10, 1);
$add1= random(1, 10, 1);
$add2= random(1, 10, 1);
$b=$a+$add1;
$c=$b+$add2;

BEGIN_TEXT
\[ \int_{$a}^{$c} f(x) - \int_{$a}^{$b} f(x) = \int_{a}^{b} f(x)\]
$BR where \( a= \)
\{ans_rule( 10)\}
and  \( b= \) \{ans_rule( 10)\}
END_TEXT

ANS(num_cmp($b), num_cmp($c));

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

##DESCRIPTION
##  Write a difference of two integrals as a single integral
##This is setIntegrals0Theory/nttheo1.pg slightly cleaned up and expanded
##by Zig Fiedorowicz, Jan. 2001
##ENDDESCRIPTION
##KEYWORDS('integrals', 'theory')

## Shotwell cleaned
## lcao , PAID on 11-24-2003

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('46')

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

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

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

$a= random(-10, 10, 1);
$a1 = random(1, 10, 1);
$a2 = random(1, 10, 1);
$a3 = random(1, 10, 1);
$b1 = random(1,3,.5);
$b = $a+$b1;
$c = $b+$b1;
$d = $c+$b1;

BEGIN_TEXT
Let \( \displaystyle \int_{$a}^{$d} f(x) dx =$a1, \ \int_{$a}^{$b} f(x) dx=$a2, \ \int_{$c}^{$d} f(x)dx =$a3 \). 
$BR$BR
Find \( \displaystyle \int_{$b}^{$c} f(x)dx= \) \{ans_rule( 20)\} $BR
$BR and \( \displaystyle \int_{$c}^{$b} ($a1 f(x)- $a2)dx= \) \{ans_rule( 20)\}

END_TEXT

$ans1="$a1-($a2)-($a3)";
$ans2="-($a1*($ans1))+$a2*$b1";

ANS(num_cmp($ans1), num_cmp($ans2));

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
This is similar to Problems 9-12 in Section 5.3 of the text.
END_TEXT
}
}

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

##DESCRIPTION
##Ellis & Gulick section 5.2 - Recognising Riemann sums
##Authored by Zig Fiedorowicz 1/12/2000
##ENDDESCRIPTION
##KEYWORDS('integrals', 'theory', 'Riemann sums')

## Shotwell cleaned
## lcao , PAID on 11-24-2003

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Definite Integral')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.2')
## Problem1('64')

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

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

$showPartialCorrectAnswers = 1;


$a = random(2,7,1);
$b = 2*$a;
$c = 3*$a;

TEXT(beginproblem());
BEGIN_TEXT
The following sum
\[\frac{1}{1+\frac{$a}{n}} \cdot \frac{$a}{n} + \frac{1}{1+\frac{$b}{n}} \cdot \frac{$a}{n}
+ \frac{1}{1+\frac{$c}{n}} \cdot \frac{$a}{n} + \ldots + \frac{1}{1+\frac{$a n}{n}} \cdot 
\frac{$a}{n}\] $BR
is a right Riemann sum for a certain definite integral
\[\int_1^b f(x)\, dx\] $BR
using a partition of the interval \([1,b]\) into \(n\) subintervals of equal length.
$BR$BR
Then the upper limit of integration must be: \(b\) = \{ ans_rule(20)\}
$BR
and the integrand must be the function \(f(x)\) =  \{ ans_rule()\}

END_TEXT

ANS(num_cmp("1+$a"));
ANS(fun_cmp("1/x"));

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
This is an easier (less abstract) version of Problem 33 in Section 5.3 of
the text. Recognizing Riemann sums is very important in applications of
integral calculus to science and engineering.
END_TEXT
}
}

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