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/11.2.5.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('5')
   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 $context = Context();
   20 
   21 
   22 ($a, $b, $s) = @{ list_random(
   23             [3, 4, 4],
   24             [5, 10, 9],
   25 ) };
   26 
   27 $sup = random(3,5);
   28 $sub = random(1,2);
   29 
   30 $s2 = 2 * $s;
   31 $a2 = 2 * $a;
   32 $b3 = 3 * $b;
   33 $a2sq = $a2 ** 2;
   34 $b3sq = $b3 ** 2;
   35 $coeff = Formula("2 * $a");
   36 $up = 1 + $s * $sup**2;
   37 $low = 1 + $s * $sub**2;
   38 
   39 #$ans = Formula("$a2/$s2 * 2/3 * ($up**(3/2) - $low**(3/2))");
   40 $ans = Compute("(2*$a2)/(3*$s2)*($up**(3/2) - $low**(3/2))");
   41 
   42 
   43 Context()->texStrings;
   44 BEGIN_TEXT
   45 \{ beginproblem() \}
   46 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","5") \}
   47 $PAR
   48 Use equation 4 to calculate the length of the path over the given interval.
   49 \[ c(t) = ($a t^2, $b t^3), \, $sub \le t \le $sup \]
   50 $PAR
   51 \{ ans_rule() \}
   52 $PAR
   53 END_TEXT
   54 Context()->normalStrings;
   55 
   56 ANS($ans->cmp);
   57 
   58 Context()->texStrings;
   59 SOLUTION(EV3(<<'END_SOLUTION'));
   60 $PAR
   61 $SOL
   62 We have \( x = $a t^2 \) and \( y = $b t^3 \).  Hence \( x' = $a2 t \) and \( y' = $b3 t^2 \).  By the formula for the arc length we get
   63 \[ S = \int ^{$sup} _{$sub} \sqrt{x'(t)^2 + y'(t)^2} \, dt = \int ^{$sup} _{$sub} \sqrt{$a2sq t^2 + $b3sq t^4} \, dt = $coeff \int ^{$sup} _{$sub} \sqrt{1 + $s t^2} \, t \, dt \]
   64 Using the substitution \( u = 1 + $s t^2, \, du = $s2 t \, dt \) we obtain
   65 \[ S = \frac{$a2}{$s2} \int ^{$up} _{$low} \sqrt{u} \, du = \frac{$a2}{$s2} \cdot \frac{2}{3} u^{3/2} \mid ^{$up} _{$low} = \frac{\{2*$a2\}}{\{3*$s2\}} (($up)^{3/2} - ($low)^{3/2}) = $ans \]
   66 END_SOLUTION
   67 
   68 ENDDOCUMENT();