## DESCRIPTION ## Calculus ## ENDDESCRIPTION ## KEYWORDS('integral' 'volume') ## Tagged by tda2d ## DBsubject('Calculus') ## DBchapter('Applications of Integration') ## DBsection('Volumes') ## Date('') ## Author('') ## Institution('Rochester') ## TitleText1('') ## EditionText1('') ## AuthorText1('') ## Section1('') ## Problem1('') 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 $BR \{image("osu_in_20_4.gif", width=>249, height=>122)\}$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 $$y$$-axis are squares. (See picture above, click for a better view.) $BR Use the formula $V=\int_a^b A(y)\,dy$ to find the volume of the formula.$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 $$y$$: \{ ans_rule(40)\}$BR $$A(y)$$= \{ 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 $PAR This problem is similar to problems 29-34 of section 8.1 of the text. END_TEXT } } ANS(num_cmp(0)); ANS(num_cmp($a2)); ANS(fun_cmp("2*sqrt(y)/$bb", vars=>"y")); ANS(fun_cmp("(2*sqrt(y)/$bb)^2", vars=>"y")); ANS(num_cmp(2*$aa**4/$b2)); ENDDOCUMENT();