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Tue Nov 8 15:17:41 2011 UTC (18 months, 1 week ago) by aubreyja
File size: 3030 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('23')
10 ## Author('Christopher Sira')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
19 $context = Context(); 20 21 sub getans 22 { 23 my($end, $num) = @_; 24$e = 2.718281828459045235360287;
25   $delx = ($num * 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   $e = 2.718281828459045235360287; 39$f = ( (sin($c))**2 + ((cos($c))**2 * $e**(2*(sin($c)))) )**(.5);
40   return $f; 41 } 42 43$a = random(2, 6);
44
45 $a10 = getans(10,$a);
46 $a20 = getans(20,$a);
47 $a30 = getans(30,$a);
48 $a50 = getans(50,$a);
49
50 $ans10 = Formula("$a10");
51 $ans20 = Formula("$a20");
52 $ans30 = Formula("$a30");
53 $ans50 = Formula("$a50");
54
55 Context()->texStrings;
56 BEGIN_TEXT
57 \{ beginproblem() \}
58 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","23") \}
59 Use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate the given curve's length.
60 $c(t) = (\cos t, e^{\sin t}) \, \, for \, 0 \le t \le a \pi$
61 $PAR 62 N = 10: \{ ans_rule() \} 63$BR
64 N = 20: \{ ans_rule() \}
65 $BR 66 N = 30: \{ ans_rule() \} 67$BR
68 N = 50: \{ ans_rule() \}
69 $PAR 70 END_TEXT 71 Context()->normalStrings; 72 73 ANS($ans10->cmp);
74 ANS($ans20->cmp); 75 ANS($ans30->cmp);
76 ANS($ans50->cmp); 77 78 Context()->texStrings; 79 SOLUTION(EV3(<<'END_SOLUTION')); 80$PAR
81 \$SOL
82 The length of the curve is given by the following integral:
83 $S = \int ^{a \pi} _0 \sqrt{ x'(t)^2 + y'(t)^2 } \, dt = \int ^{a \pi} _0 \sqrt{ \left( - \sin t \right)^2 + \left( \cos t e^{\sin t} \right)^2 } \, dt$
84 That is, $$S = \int ^{a \pi} _0 \sqrt{\sin ^2 t + \cos^2 t e^{2 \sin t} } \, dt$$.  We approximate the integral using the Midpoint Rule with N = 10, 20, 30, 50.  For $$f(t) = \sqrt{ \sin^2 t + \cos^2 t e^{2 \sin t} }$$ we obtain
85 $(N = 10): \Delta x = \frac{a \pi}{10}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{a \pi}{10}$
86 $M_{10} = \frac{a \pi}{10} \sum ^{10} _{i=1} f(c_i) = ans10$
87 $(N = 20): \Delta x = \frac{a \pi}{20}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{a \pi}{20}$
88 $M_{20} = \frac{a \pi}{20} \sum ^{20} _{i=1} f(c_i) = ans20$
89 $(N = 30): \Delta x = \frac{a \pi}{30}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{a \pi}{30}$
90 $M_{30} = \frac{a \pi}{30} \sum ^{30} _{i=1} f(c_i) = ans30$
91 $(N = 50): \Delta x = \frac{a \pi}{50}, \, c_i = \left( i - \frac{1}{2} \right) \cdot \frac{a \pi}{50}$
92 $M_{50} = \frac{a \pi}{50} \sum ^{50} _{i=1} f(c_i) = ans50$
93 END_SOLUTION
94
95 ENDDOCUMENT();