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| 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(); |
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