#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(<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();