## 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(<"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.