View of /trunk/NationalProblemLibrary/Union/setIntSurfaceArea/ur_in_27_4.pg

    1 ## DESCRIPTION
    2 ##   Calculate a Surface Area of a Solid of Revolution
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('Surface Area', 'Solid of Revolution', 'Rotate Curve')
    6 ## Tagged by nhamblet
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Applications of Integration')
   10 ## DBsection('Solids of Revolution')
   11 ## Date('8/23/07')
   12 ## Author('')
   13 ## Institution('Union')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 ## TitleText2('Calculus: Early Transcendentals')
   20 ## EditionText2('1')
   21 ## AuthorText2('Rogawski')
   22 ## Section2('6.3')
   23 ## Problem2('11')
   24 
   25 DOCUMENT();        # This should be the first executable line in the problem.
   26 
   27 loadMacros(
   28   "PGstandard.pl",
   29   "PGunion.pl",            # Union College utilities
   30   "MathObjects.pl",
   31   "PGcourse.pl",           # Customization file for the course
   32 );
   33 
   34 TEXT(beginproblem());
   35 
   36 ###################################
   37 # Setup
   38 
   39 $a = random(2,6,1);
   40 $b = random(1,9,1);
   41 
   42 Context()->variables->add(y=>'Real');
   43 $curve=Formula("$a e^{2y}");
   44 $left=0;
   45 $right=$b;
   46 
   47 ###################################
   48 # Main text
   49 
   50 Context()->texStrings;
   51 BEGIN_TEXT
   52 
   53 Find the area of the surface obtained by rotating the curve
   54 \[x = $curve\]
   55 from \(y = $left\) to \(y= $right\) about the \(y\)-axis.
   56 $PAR
   57 
   58 The area is \{ans_rule(50)\} square units.
   59 
   60 END_TEXT
   61 Context()->normalStrings;
   62 
   63 ###################################
   64 # Answers
   65 
   66 $showPartialCorrectAnswers = 1;
   67 Context()->flags->set(tolerance => .001, reduceConstants=>0, reduceConstantFunctions => 0);
   68 
   69 ## In this problem, need to integrate 2pi * ae^(2y) * sqrt(1+(2ae^(2y))^2 ).
   70 ## Do this by 'u-sub': u=2ae^(2y) so then need to integrate pi/2 * sqrt(1+u^2) ...
   71 
   72 $u = "2*$a*exp(2*$b)";
   73 $v = "2*$a";
   74 $c = "sqrt(1+{$u}**2)";
   75 $d = "sqrt(1+{$v}**2)";
   76 
   77 $answer = "(pi/4) * (($u*$c+ln($u+$c)) - ($v*$d+ln($v+$d)))";
   78 
   79 ANS(Compute($answer)->cmp);
   80 
   81 ###################################
   82 
   83 
   84 COMMENT('MathObject version');
   85 ENDDOCUMENT();