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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('25')
   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 $context = Context();
   19 $context->flags->set(tolerance=>'0.00005', tolType=>'absolute');
   20 
   21 sub getans
   22 {
   23   my($end) = @_;
   24   $e = 2.718281828459045235360287;
   25   $delx = (2 * 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   $f = ( $bsq + $diff*(sin($c))**2 )**(.5);
   39   return $f;
   40 }
   41 
   42 $a = random(5, 8);
   43 $b = random(2, 4);
   44 $asq = $a**2;
   45 $bsq = $b**2;
   46 $diff = $asq - $bsq;
   47 
   48 $a10 = getans(10);
   49 $a20 = getans(20);
   50 $a30 = getans(30);
   51 $a50 = getans(50);
   52 
   53 $ans10 = Formula("$a10");
   54 $ans20 = Formula("$a20");
   55 $ans30 = Formula("$a30");
   56 $ans50 = Formula("$a50");
   57 
   58 Context()->texStrings;
   59 BEGIN_TEXT
   60 \{ beginproblem() \}
   61 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","25") \}
   62 Use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate the given curve's length.
   63 The ellipse \( \left( \frac{x}{$a} \right) ^2 + \left( \frac{y}{$b} \right) ^2 = 1 \)
   64 $PAR
   65 N = 10: \{ ans_rule() \}
   66 $BR
   67 N = 20: \{ ans_rule() \}
   68 $BR
   69 N = 30: \{ ans_rule() \}
   70 $BR
   71 N = 50: \{ ans_rule() \}
   72 $PAR
   73 END_TEXT
   74 Context()->normalStrings;
   75 
   76 ANS($ans10->cmp);
   77 ANS($ans20->cmp);
   78 ANS($ans30->cmp);
   79 ANS($ans50->cmp);
   80 
   81 Context()->texStrings;
   82 SOLUTION(EV3(<<'END_SOLUTION'));
   83 $PAR
   84 $SOL
   85 We use the parameterization given in Example 4, section 12.1, that is, \( c(t) = ($a \cos t, \, $b \sin t), \, 0 \le t \le 2 \pi \).
   86 The length of the curve is given by the following integral:
   87 \[ S = \int ^{2 \pi} _0 \sqrt{ x'(t)^2 + y'(t)^2 } \, dt = \int ^{2 \pi} _0 \sqrt{ \left( - $a \sin t \right)^2 + \left( $b \cos t \right)^2 } \, dt \]
   88 \[ = \int ^{2 \pi} _0 \sqrt{ $asq \sin^2 t + $bsq \cos^2t } \, dt = \int ^{2 \pi} _0 \sqrt{ $bsq \left( \sin^2 t + \cos^2 t \right) + $diff \sin^2 t } \, dt = \int ^{2 \pi} _0 \sqrt{ $bsq + $diff \sin^2 t } \, dt \]
   89 That is,
   90 \[ S = \int ^{2 \pi} _0 \sqrt{ $bsq + $diff \sin^2 t } \, dt \]
   91 We approximate the integral using the Midpoint Rule with N = 10, 20, 30, 50, for \( f(t) = \sqrt{ $bsq + $diff \sin^2 t } \).  We obtain
   92 
   93 \[ (N = 10): \Delta x = \frac{2 \pi}{10}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{\pi}{5} \]
   94 \[ M_{10} = \frac{\pi}{5} \sum ^{10} _{i=1} f(c_i) = $ans10 \]
   95 \[ (N = 20): \Delta x = \frac{2 \pi}{20}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{\pi}{10} \]
   96 \[ M_{20} = \frac{\pi}{10} \sum ^{20} _{i=1} f(c_i) = $ans20 \]
   97 \[ (N = 30): \Delta x = \frac{2 \pi}{30}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{\pi}{15} \]
   98 \[ M_{30} = \frac{\pi}{15} \sum ^{30} _{i=1} f(c_i) = $ans30 \]
   99 \[ (N = 50): \Delta x = \frac{2 \pi}{50}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{\pi}{25} \]
  100 \[ M_{50} = \frac{\pi}{25} \sum ^{50} _{i=1} f(c_i) = $ans50 \]
  101 END_SOLUTION
  102 
  103 ENDDOCUMENT();