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Tue Nov 8 15:17:41 2011 UTC (18 months, 2 weeks ago) by aubreyja
File size: 1838 byte(s)
Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    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();
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();


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