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| 1 : | aubreyja | 2584 | ## DBsubject('Calculus') |
| 2 : | ## DBchapter('Techniques of Integration') | ||
| 3 : | ## DBsection('Trigonometric Substitution') | ||
| 4 : | ## KEYWORDS('calculus', 'integration', 'integral', 'trigonometric substitution', 'substitution', 'trigonometry', 'trigonometric', 'trig') | ||
| 5 : | ## TitleText1('Calculus: Early Transcendentals') | ||
| 6 : | ## EditionText1('2') | ||
| 7 : | ## AuthorText1('Rogawski') | ||
| 8 : | ## Section1('7.3') | ||
| 9 : | ## Problem1('58') | ||
| 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 = 0; | ||
| 21 : | # Introduced more variation and kept same level of difficulty | ||
| 22 : | $c = random(2, 4, 1); | ||
| 23 : | $d = random(5, 9, 1); | ||
| 24 : | $b = $c*$c; | ||
| 25 : | $d2 = 2*$d; | ||
| 26 : | #$dc4 = $d *($c**4)/4; | ||
| 27 : | $dc4 = $d *($c**4)/8; | ||
| 28 : | $ans = Formula(" $dc4 * (pi ** 2) ")->reduce(); | ||
| 29 : | |||
| 30 : | Context()->texStrings; | ||
| 31 : | BEGIN_TEXT | ||
| 32 : | \{ beginproblem() \} | ||
| 33 : | \{ textbook_ref_exact("Rogawski ET 2e", "7.3","58") \} | ||
| 34 : | $PAR | ||
| 35 : | Find the volume of the solid obtained by revolving the graph of \( y = $d x\sqrt{$b - x^2} \) | ||
| 36 : | over [$a,$c] about the y-axis. | ||
| 37 : | $PAR | ||
| 38 : | \{ans_rule()\} | ||
| 39 : | $PAR | ||
| 40 : | END_TEXT | ||
| 41 : | Context()->normalStrings; | ||
| 42 : | |||
| 43 : | ANS($ans->cmp); | ||
| 44 : | |||
| 45 : | Context()->texStrings; | ||
| 46 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 47 : | $PAR | ||
| 48 : | $SOL | ||
| 49 : | $PAR | ||
| 50 : | Using the method of cylindrical shells, the volume is given by | ||
| 51 : | $PAR | ||
| 52 : | \( V = 2\pi \int^{$c}_{$a} x ($d x\sqrt{$b - x^2}) \, dx = $d2 \pi \int^{$c}_{$a} x^2 \sqrt{$b - x^2} \, dx \). | ||
| 53 : | $PAR | ||
| 54 : | To evaluate this integral, let \( x = $c \sin \theta \). Then \( dx = $c \cos \theta \, d\theta \), | ||
| 55 : | $PAR | ||
| 56 : | \( $b - x^2 = $b (1 - \sin^2 \theta) = $b \cos^2 \theta \), | ||
| 57 : | $PAR | ||
| 58 : | and | ||
| 59 : | \[ | ||
| 60 : | \begin{array}{ll} | ||
| 61 : | I & = \int x^2\sqrt{$b - x^2} \, dx \cr | ||
| 62 : | & = \int ($c^2 \sin^2 \theta )\,( $c \cos \theta )\, ($c \cos \theta) \, d\theta \cr | ||
| 63 : | & = $c^{4} \int (1 - \cos^2 \theta) \cos^2 \theta \, d\theta. \cr | ||
| 64 : | & \cr | ||
| 65 : | \frac{1}{$c^{4}} I &= \int \cos^2 \theta \, d\theta - \int \cos^4 \theta \, d\theta . | ||
| 66 : | \end{array} | ||
| 67 : | \] | ||
| 68 : | |||
| 69 : | Now we use the reduction formula for \( \int \cos^4 \theta \, d\theta \): | ||
| 70 : | \[ | ||
| 71 : | \begin{array}{ll} | ||
| 72 : | \frac{1}{$c^{4}} I & = \int \cos^2 \theta \, d\theta - \left[ \frac{cos^3 \theta \, \sin \theta}{4} + \frac{3}{4}\int \cos^2 \theta \, d\theta \right] \cr | ||
| 73 : | & = -\frac{1}{4}\cos^3 \theta \, \sin \theta + \frac{1}{4} \int \cos^2 \theta \, d\theta \cr | ||
| 74 : | & = -\frac{1}{4}\cos^3 \theta \, \sin \theta + \frac{1}{4} \left[ \frac{1}{2} \theta + \frac{1}{2} \sin \theta \, \cos \theta \right] + C \cr | ||
| 75 : | & = -\frac{1}{4}\cos^3 \theta \, \sin \theta + \frac{1}{8} \theta + \frac{1}{8} \sin \theta \, \cos \theta + C. | ||
| 76 : | \end{array} | ||
| 77 : | \] | ||
| 78 : | |||
| 79 : | $PAR | ||
| 80 : | Since \( \sin \theta = \frac{x}{$c} \), we know that \( \cos \theta = \frac{\sqrt{$b - x^2} }{$c}\). | ||
| 81 : | Then we have | ||
| 82 : | $PAR | ||
| 83 : | \(\frac{1}{$c^{4}} I = -\frac{1}{4}\left(\frac{\sqrt{$b - x^2} }{$c}\right)^3 \, \left(\frac{x}{$c}\right) + \frac{1}{8}\sin^{-1} \left(\frac{x}{$c}\right) | ||
| 84 : | + \frac{1}{8}\left(\frac{x}{$c}\right) \left(\frac{\sqrt{$b - x^2} }{$c}\right) + C \). | ||
| 85 : | $PAR | ||
| 86 : | Now we can complete the volume calculation: | ||
| 87 : | \[ | ||
| 88 : | \begin{array}{ll} | ||
| 89 : | V & = \left . $d2 \pi I \right |^{$c}_{$a} \cr | ||
| 90 : | & = $d2 \cdot $c^{4} \pi \left [ -\frac{1}{4}\left(\frac{\sqrt{$b - x^2} }{$c}\right)^3 \, \left(\frac{x}{$c}\right) + \frac{1}{8}\sin^{-1} \left(\frac{x}{$c}\right) \right .\cr | ||
| 91 : | & \left . \quad + \frac{1}{8}\left(\frac{x}{$c}\right) \left(\frac{\sqrt{$b - x^2} }{$c}\right) \right ]^{$c}_{$a} \cr | ||
| 92 : | & = $d2 \cdot $c^{4} \pi \frac{1}{8} \frac{\pi}{2} = $d \cdot $c^{4} \pi^2 \frac{1}{4} \cr | ||
| 93 : | & = $ans . | ||
| 94 : | \end{array} | ||
| 95 : | \] | ||
| 96 : | $PAR | ||
| 97 : | END_SOLUTION | ||
| 98 : | |||
| 99 : | ENDDOCUMENT(); | ||
| 100 : | |||
| 101 : |
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