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# Annotation of /trunk/NationalProblemLibrary/Rochester/setIntegrals20Volume/osu_in_20_4/osu_in_20_4.pg

 1 : jj 144 ##DESCRIPTION 2 : ##KEYWORDS('integrals', 'volume') 3 : ##Ellis and Gullick: section 8.1 4 : ##Authored by Zig Fiedorowicz 5/19/2000 5 : ##ENDDESCRIPTION 6 : 7 : DOCUMENT(); 8 : 9 : loadMacros( 10 : "PG.pl", 11 : "PGbasicmacros.pl", 12 : "PGchoicemacros.pl", 13 : "PGanswermacros.pl", 14 : "PGauxiliaryFunctions.pl", 15 : "PGgraders.pl" 16 : ); 17 : 18 : ##Note this uses Mike Gage's custom full_partial_grader 19 : ##contained in file PGgraders.pl 20 : install_problem_grader(~~&full_partial_grader); 21 : 22 : $showPartialCorrectAnswers = 1; 23 : 24 :$aa = random(2,6); 25 : $a2 =$aa*$aa; 26 :$bb = random(2,6); 27 : if ($bb==$aa) {$bb++;} 28 :$b2 = $bb*$bb; 29 : 30 : TEXT(beginproblem()); 31 : BEGIN_TEXT 32 : $BR 33 : 34 : \{image("osu_in_20_4.gif", width=>249, height=>122)\} 35 : 36 :$BR 37 : The base of a certain solid is the area bounded above by the graph of $$y=f(x)=a2$$ 38 : and below by the graph of $$y=g(x)= b2 x^2$$. Cross-sections perpendicular to the $$y$$-axis 39 : are squares. (See picture above, click for a better view.) 40 : $BR 41 : Use the formula 42 : $V=\int_a^b A(y)\,dy$ 43 : to find the volume of the formula. 44 :$BR 45 : {\bf Note:} You can get full credit for this problem by just entering the final 46 : answer (to the last question) correctly. The initial questions are meant as hints 47 : towards the final answer and also allow you the opportunity to get partial credit. 48 : $BR 49 : The lower limit of integration is $$a$$ = \{ ans_rule()\} 50 :$BR 51 : 52 : The upper limit of integration is $$b$$ = \{ ans_rule()\} 53 : $BR 54 : 55 : The side $$s$$ of the square cross-section is the following function of $$y$$: 56 : \{ ans_rule(40)\} 57 :$BR 58 : 59 : $$A(y)$$= \{ ans_rule(40)\} 60 : $BR 61 : Thus the volume of the solid is $$V$$ = \{ ans_rule()\} 62 : END_TEXT 63 : 64 : ##set$PG_environment{'textbook'} in webworkCourse.ph 65 : if (defined($textbook)) { 66 : if ($textbook eq "EllisGulick5") { 67 : BEGIN_TEXT 68 : $PAR 69 : This problem is similar to problems 29-34 of section 8.1 of the text. 70 : END_TEXT 71 : } 72 : } 73 : ANS(num_cmp(0)); 74 : ANS(num_cmp($a2)); 75 : ANS(fun_cmp("2*sqrt(y)/$bb", vars=>"y")); 76 : ANS(fun_cmp("(2*sqrt(y)/$bb)^2", vars=>"y")); 77 : ANS(num_cmp(2*$aa**4/$b2)); 78 : 79 : ENDDOCUMENT();