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    1 ## DBsubject('Calculus')
    2 ## DBchapter('Parametric Equations, Polar Coordinates, and Conic Sections')
    3 ## DBsection('Arc Length and Speed')
    4 ## KEYWORDS('calculus', 'parametric', 'polar', 'conic')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('11.2')
    9 ## Problem1('23')
   10 ## Author('Christopher Sira')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("PGchoicemacros.pl");
   16 loadMacros("Parser.pl");
   17 loadMacros("freemanMacros.pl");
   18 loadMacros("PGchoicemacros.pl");
   19 $context = Context();
   20 
   21 sub getans
   22 {
   23   my($end, $num) = @_;
   24   $e = 2.718281828459045235360287;
   25   $delx = ($num * pi) / $end;
   26   $ans = 0;
   27   for($i=1; $i<=$end; $i++)
   28   {
   29     $ci = ($i - .5) * $delx;
   30     $ans += func($ci);
   31   }
   32   return $delx * $ans;
   33 }
   34 
   35 sub func
   36 {
   37   my($c) = @_;
   38   $e = 2.718281828459045235360287;
   39   $f = ( (sin($c))**2 + ((cos($c))**2 * $e**(2*(sin($c)))) )**(.5);
   40   return $f;
   41 }
   42 
   43 $a = random(2, 6);
   44 
   45 $a10 = getans(10, $a);
   46 $a20 = getans(20, $a);
   47 $a30 = getans(30, $a);
   48 $a50 = getans(50, $a);
   49 
   50 $ans10 = Formula("$a10");
   51 $ans20 = Formula("$a20");
   52 $ans30 = Formula("$a30");
   53 $ans50 = Formula("$a50");
   54 
   55 Context()->texStrings;
   56 BEGIN_TEXT
   57 \{ beginproblem() \}
   58 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","23") \}
   59 Use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate the given curve's length.
   60 \[ c(t) = (\cos t, e^{\sin t}) \, \, for \, 0 \le t \le $a \pi \]
   61 $PAR
   62 N = 10: \{ ans_rule() \}
   63 $BR
   64 N = 20: \{ ans_rule() \}
   65 $BR
   66 N = 30: \{ ans_rule() \}
   67 $BR
   68 N = 50: \{ ans_rule() \}
   69 $PAR
   70 END_TEXT
   71 Context()->normalStrings;
   72 
   73 ANS($ans10->cmp);
   74 ANS($ans20->cmp);
   75 ANS($ans30->cmp);
   76 ANS($ans50->cmp);
   77 
   78 Context()->texStrings;
   79 SOLUTION(EV3(<<'END_SOLUTION'));
   80 $PAR
   81 $SOL
   82 The length of the curve is given by the following integral:
   83 \[ S = \int ^{$a \pi} _0 \sqrt{ x'(t)^2 + y'(t)^2 } \, dt = \int ^{$a \pi} _0 \sqrt{ \left( - \sin t \right)^2 + \left( \cos t e^{\sin t} \right)^2 } \, dt \]
   84 That is, \( S = \int ^{$a \pi} _0 \sqrt{\sin ^2 t + \cos^2 t e^{2 \sin t} } \, dt \).  We approximate the integral using the Midpoint Rule with N = 10, 20, 30, 50.  For \( f(t) = \sqrt{ \sin^2 t + \cos^2 t e^{2 \sin t} } \) we obtain
   85 \[ (N = 10): \Delta x = \frac{$a \pi}{10}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{$a \pi}{10} \]
   86 \[ M_{10} = \frac{$a \pi}{10} \sum ^{10} _{i=1} f(c_i) = $ans10 \]
   87 \[ (N = 20): \Delta x = \frac{$a \pi}{20}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{$a \pi}{20} \]
   88 \[ M_{20} = \frac{$a \pi}{20} \sum ^{20} _{i=1} f(c_i) = $ans20 \]
   89 \[ (N = 30): \Delta x = \frac{$a \pi}{30}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{$a \pi}{30} \]
   90 \[ M_{30} = \frac{$a \pi}{30} \sum ^{30} _{i=1} f(c_i) = $ans30 \]
   91 \[ (N = 50): \Delta x = \frac{$a \pi}{50}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{$a \pi}{50} \]
   92 \[ M_{50} = \frac{$a \pi}{50} \sum ^{50} _{i=1} f(c_i) = $ans50 \]
   93 END_SOLUTION
   94 
   95 ENDDOCUMENT();