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/11.2.32.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('32')
   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 ## DBsubject('Calculus')
   21 ## DBchapter('Parametric Equations, Polar Coordinates, and Conic Sections')
   22 ## DBsection('Arc Length and Speed')
   23 ## KEYWORDS('calculus', 'parametric', 'polar', 'conic')
   24 ## TitleText1('Calculus: Early Transcendentals 2e')
   25 ## EditionText1('2')
   26 ## AuthorText1('Rogawski')
   27 ## Section1('11.2')
   28 ## Problem1('31')
   29 ## Author('Christopher Sira')
   30 ## Institution('W.H.Freeman')
   31 
   32 DOCUMENT();
   33 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   34 loadMacros("PGchoicemacros.pl");
   35 loadMacros("Parser.pl");
   36 loadMacros("freemanMacros.pl");
   37 $context = Context();
   38 
   39 
   40 @list = qw(\frac{\pi}{6} \frac{\pi}{4} \frac{\pi}{3} \frac{\pi}{2});
   41 @upps = qw(\frac{1}{2} \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} 1);
   42 @anss = (Compute("6 * pi * ((1/2)**5) / 5"), Compute("6 * pi * ((sqrt(2)/2)**5) / 5") ,Compute("6 * pi * ((sqrt(3)/2)**5) / 5"),Compute("6 * pi  / 5"));
   43 @anspr = qw(6\pi\frac{\left(\frac{1}{2}\right)^5}{5} 6\pi\frac{\left(\frac{\sqrt{2}}{2}\right)^5}{5} 6\pi\frac{\left(\frac{\sqrt{3}}{2}\right)^5}{5} \frac{6\pi}{5});
   44 
   45 
   46 $l = random(0,3);
   47 $a = $list[$l];
   48 
   49 $upp = $upps[$l];
   50 
   51 $ans = $anss[$l];
   52 $ansp = $anspr[$l];
   53 
   54 Context()->texStrings;
   55 BEGIN_TEXT
   56 \{ beginproblem() \}
   57 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","32") \}
   58 Compute the surface area of the surface generated by revolving the astroid with parametrization \( c(t) = ( \cos^3 t, \, \sin^3 t) \) about the x-axis for \( 0 \le t \le $a \)
   59 $PAR
   60 \{ ans_rule() \}
   61 $PAR
   62 END_TEXT
   63 Context()->normalStrings;
   64 
   65 ANS($ans->cmp);
   66 
   67 Context()->texStrings;
   68 SOLUTION(EV3(<<'END_SOLUTION'));
   69 $PAR
   70 $SOL
   71 We have \( x(t) = \cos^3 t, \, y(t) = \sin^3 t, \, x'(t) = -3 \cos^2 t \sin t, \, y'(t) = 3 \sin^2 t \cos t \).  Hence,
   72 \[ x'(t)^2 + y'(t)^2 = 9 \cos^4 t \sin^2 t + 9 \sin^4 t \cos^2 t = 9 \cos^2 t \sin^2 t \left( \cos^2 t + \sin^2 t \right) = 9 \cos^2 t \sin^2 t \]
   73 Using the formula for the surface area we get
   74 \[ S = 2 \pi \int ^{$a} _0 y(t) \sqrt{ x'(t)^2 + y'(t)^2 } \, dt = 2 \pi \int ^{$a} _0 \sin^3 t \cdot 3 \cos t \sin t \, dt = 6 \pi \int ^{$a} _0 \sin^4 t \cos t \, dt \]
   75 We compute the integral using the substitution \( u = \sin t, \, du = \cos t\, dt \).  We obtain
   76 \[ S = 6 \pi \int ^{$upp} _0 u^4 \, du = 6 \pi \frac{u^5}{5} \mid ^{$upp} _0 = $ansp \approx \{$ans\} \]
   77 END_SOLUTION
   78 
   79 ENDDOCUMENT();