View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/11_Parametric_Equations_Polar_Coordinates_and_Conic_Sections/11.2_Arc_Length_and_Speed/11.2.8.pg

    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('8')
   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 
   20 $a = random(2, 9);
   21 
   22 $ans = Formula("$a**2 / 2");
   23 
   24 Context()->texStrings;
   25 BEGIN_TEXT
   26 \{ beginproblem() \}
   27 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","8") \}
   28 Use equation 4 to calculate the length of the path over the given interval.
   29 \[ (\sin \theta - \theta \cos \theta, \cos \theta + \theta \sin \theta), \, 0 \le \theta \le $a \]
   30 $PAR
   31 \{ ans_rule() \}
   32 $PAR
   33 END_TEXT
   34 Context()->normalStrings;
   35 
   36 ANS($ans->cmp);
   37 
   38 Context()->texStrings;
   39 SOLUTION(EV3(<<'END_SOLUTION'));
   40 $PAR
   41 $SOL
   42 We have \( x = \sin \theta - \theta \cos \theta \) and \( y = \cos \theta + \theta \sin \theta \).  Hence, \( x' = \cos \theta - \left( \cos \theta - \theta \sin \theta \right) = \theta \sin \theta \) and \( y' = - \sin \theta + \sin \theta + \theta \cos \theta = \theta \cos \theta \).  Using the formula for the arc length we obtain:
   43 \[ S = \int ^{$a} _0 \sqrt{x'(\theta)^2 + y'(\theta)^2} \, dt = \int ^{$a} _0 \sqrt{ \left( \theta \sin \theta \right) ^2 + \left( \theta \cos \theta \right)^2 } \, d \theta \]
   44 \[ = \int ^{$a} _0 \sqrt{ \theta^2 \left( \sin^2 \theta + \cos^2 \theta \right) } \, d \theta = \int ^{$a} _0 \theta \, d \theta = \frac{\theta^2}{2} \mid ^{$a} _0 = \frac{\{$a**2\}}{2} = $ans \]
   45 END_SOLUTION
   46 
   47 ENDDOCUMENT();