660 Chapter 10 Conic Sections and Polar Coordinates
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1 Chpter Conic Sections nd Polr Coordintes 8. ( (b (c (d (e r r Ê r ; therefore cos Ê Ê ( ß is point of intersection ˆ ˆ Ê Ê Ê ß ß ˆ ß 9. ( r cos Ê cos ; r cos Ê r Š Ê r r Ê (r (b r Ê cos Ê cos Ê, Ê ß or ß is on the grph; r Ê cos cos ( is on the grph. Since ( for polr coordintes, the grphs intersect t the origin. 5. ( Let r f( be symmetric bout the x-xis nd the y-xis. Then (r ß on the grph Ê (r ß is on the grph becuse of symmetry bout the x-xis. Then ( ß r ( ( ß r is on the grph becuse of symmetry bout the y-xis. Therefore r f( is symmetric bout the origin. (b Let r f( be symmetric bout the x-xis nd the origin. Then (r ß on the grph Ê (r ß is on the grph becuse of symmetry bout the x-xis. Then ( ß r is on the grph becuse of symmetry bout the origin. Therefore r f( is symmetric bout the y-xis. (c Let r f( be symmetric bout the y-xis nd the origin. Then (r ß on the grph Ê ( r ß is on the grph becuse of symmetry bout the y-xis. Then ( ( r ß (r ß is on the grph becuse of symmetry bout the origin. Therefore r f( is symmetric bout the x-xis. 5. The mximum width of the petl of the rose which lies long the x-xis is twice the lrgest y vlue of the curve on the intervl Ÿ Ÿ. So we wish to mximize y r sin cos sin on Ÿ Ÿ. Let $ w f( cos sin sin b(sin sin sin Ê f ( cos sin cos. Then f w ( cos sin cos (cos sin cos or sin Ê Ê b Ê Ê or $ Š 9 Ÿ Ÿ Š 9 sin. Since we wnt Ÿ Ÿ, we choose sin Š Ê f( sin sin. We cn see from the grph of r cos tht mximum does occur in the intervl. Therefore the mximum width occurs t sin, nd the mximum width is. 5. We wish to mximize y r sin ( cos (sin sin sin cos. Then dy d cos (sin ( sin cos cos cos sin cos cos cos ; thus dy d Ê cos cos = Ê cos cos Ê ( cos (cos Ê cos or cos Ê 5,,. From the grph, we cn see tht the mximum occurs in the first qunt so we choose. Then y sin sin cos. The x-coordinte of this point is x r cos ˆ cos ˆ cos. Thus the mximum height is h occurring t x.
2 .7 AREA AND LENGTHS IN POLAR COORDINATES Section.7 Are nd Lengths in Polr Coordintes. A ( cos d cos cos b d 8 8 cos ˆ cos d (9 8 cos cos d 9 8 sin sin 8. A [( cos ] d cos cos d cos d b ˆ cos ˆ cos cos d sin sin % 8 cos. A cos d d sin. A cos b d cos d sin % % c c 5. A ( sin d sin d ccos d. A (( ( sin d sin d 7. r cos nd r sin Ê cos sin Ê cos sin Ê ; therefore ˆ cos d % A ( sin d sin d d ( cos d c sin cos 8. r nd r sin Ê sin Ê sin 5 Ê or ; therefore 5 c A ( ( sin d d 5 ˆ sin d 5 ˆ cos d sin & 5 ˆ cos d ˆ 5 5 sin ˆ sin
3 Chpter Conic Sections nd Polr Coordintes 9. r nd r ( cos Ê ( cos Ê cos Ê ; therefore cos cos b d ˆ ( ˆ cos cos d A [( cos ] d re of the circle ( 8 cos cos d c 8 sin sin d 58. r ( cos nd r ( cos Ê cos cos Ê cos Ê or ; the grph lso gives the point of intersection (ß ; therefore A [( cos ] d [( cos ] d cos cos bd cos cos b d ˆ cos cos d ˆ cos cos d ( 8 cos cos d ( 8 cos cos d c 8 sin sin d c 8 sin sin d. r nd r cos Ê cos Ê cos Ê (in the st qunt; we use symmetry of the grph to find the re, so A ( cos Š d ( cos d c sin d. r cos nd r ( cos Ê cos ( cos Ê cos cos Ê cos Ê or ; the grph lso gives the point of intersection (ß ; therefore A c( cos ( cos dd 9 cos cos cos d b 8 cos cos b d c ( cos cos d d cos cos b d $ c sin sin d ˆ Š Š
4 . r nd r cos Ê cos Ê cos Ê in qunt II; therefore A c( cos d d cos b d [( cos ] d ( cos d sin d $ c Section.7 Are nd Lengths in Polr Coordintes. ( A ( cos d cos cos bd [( cos cos ] d ( cos cos d c sin sin d $ (b A Š Š (from ( bove nd Exmple in the text 5. r nd r csc Ê sin Ê sin 5 5 Ê or ; therefore A 9 csc b d 5 ˆ 9 8 csc d 9 8 cot & 9 Š 5 9 Š r cos nd r sec Ê sec cos Ê cos cos Ê 8 cos b cos b % % % Ê cos cos Ê cos cos Ê cos 8 cos 8 8 Ê sin tn Ê cos b cos b Ê cos or cos Ê cos (the second eqution hs no rel roots (in the first qunt; thus A ˆ cos sec d ˆ cos sec d Š 7. ( r tn nd r Š csc Ê tn Š csc Ê sin Š cos Ê cos Š cos Ê cos Š cos Ê cos or (use the qutic formul Ê (the solution in the first qunt; therefore the re of R is A tn d sec b d ctn d ˆ tn ; % 8 AO Š csc nd OB Š csc Ê AB Ê Š Ê the re of R is A Š Š ; therefore the re of the region shded in the text is ˆ. Note: The re must be found this wy since no common intervl genertes the region. For 8 exmple, the intervl Ÿ Ÿ genertes the rc OB of r tn but does not generte the segment AB of the line r csc. Insted the intervl genertes the hlf-line from B to _ on the line r csc. (b lim tn _ nd the line x is r sec in polr coordintes; then lim (tn sec Ä Ä c = lim ˆ sin lim ˆ sin lim ˆ cos r tn pproches Ä c cos cos Ä c cos Ä c sin Ê r sec s Ä Ê r sec (or x is verticl symptote of r tn. Similrly, r sec c
5 Chpter Conic Sections nd Polr Coordintes (or x is verticl symptote of r tn. 8. It is not becuse the circle is generted twice from to. The re of the crdioid is A (cos d cos cos bd ˆ cos cos d sin ˆ 5 sin. The re of the circle is A Ê the re requested is ctully r, Ÿ Ÿ 5 Ê ; therefore Length É b ( d % d d 5 5 kk d (since d ; u Ê du d ; Ê u, 9 5 Ê u 9 Ä u du u $ * 9 % e e e e e d. r, Ÿ Ÿ Ê ; therefore Length ÊŠ Š d Ê Š d e d e e. r cos Ê sin ; therefore Length ( cos ( sin d d ( cos É cos É ˆ cos d d d cos d sin 8. r sin, Ÿ Ÿ, Ê sin cos ; therefore Length Ɉ sin ˆ sin cos d d É % É cos sin sin cos d sin sin cos d (since Ÿ Ÿ sin ˆ d sin sin cos d (cos cos (cos. r, Ÿ Ÿ Ê ; therefore Length ʈ Š d sin sin ( cos cos b cos ( cos É d % É d ˆ since on Ÿ Ÿ ˆ cos cos sin É d cos cos (cos ˆ cos É d d d $ sec d cos ( cos ( cos $ $ ˆ cos % $ $ sec d sec u du (use tbles Œ sec u tn u sec u du Š % ln ksec u tn uk ln Š sin sin. r cos, Ÿ Ÿ Ê d ( cos ; therefore Length ʈ cos Š (cos d sin ( cos sin ( cos cos b cos ( cos Ê Š d É d ˆ since cos on Ÿ Ÿ ˆ cos cos sin É d cos (cos ˆ cos É d d d $ csc d cos ( cos ( cos $ $ ˆ sin $ csc ˆ d ˆ since csc on $ csc u du (use tbles Ÿ Ÿ
6 Section.7 Are nd Lengths in Polr Coordintes 5 Œ csc u cot u csc u du Š ln kcsc u cot uk ln Š % % lnš d É % É $ 5. r cos Ê sin cos ; therefore Length Ɉ cos $ ˆ sin cos d cos ˆ sin ˆ cos ˆ d ˆ cos cos ˆ sin ˆ d cos ˆ d cosˆ % 8 8 d sin. r sin, Ÿ Ÿ Ê d ( sin ( cos (cos ( sin ; therefore Length cos É( sin d sin sin cos É d ( sin sin sin É d d sin 7. r cos Ê ( cos ( sin ; therefore Length sin É( cos d d ( cos cos cos sin É d cos É d d cos cos 8. ( r Ê ; Length d kk d c d d (b r cos Ê sin ; Length ( cos ( sin d cos sin bd d kk d c d (c r sin Ê cos ; Length ( cos ( sin d cos sin b d d kk d c d 9. r sin cos, Ÿ Ÿ Ê (cos ( sin ( ; therefore Surfce Are d cos ( r cos ÊŠ sin cos Š d Š cos (cos É sin cos d cos cos Š % cos (cos É d cos d c sin d cos d. r e, Ÿ Ÿ Ê ˆ e e ; therefore Surfce Are Š e (sin Š e Š e d Š Ê e (sin Ée e d Š 5 e (sin e d Š 5 É e (sin Š e d 5 e sin d e b 5 (sin cos 5 e where we integrted by prts. r cos Ê r cos ; use r cos on ß sin Ê (cos ( sin ( ; d cos therefore Surfce Are Š cos (sin É sin cos d cos (sin É d cos cos % sin d ccos d ( Š
7 Chpter Conic Sections nd Polr Coordintes d b b b kk ˆ cos. r cos Ê sin ; therefore Surfce Are ( cos (cos ( cos ( sin d cos cos sin d 8 cos d 8 cos d 8 d ( cos d sin d d w w dy w d d dy w w w Š d c d c d dx w. Let r f(. Then x f( cos Ê f ( cos f( sin Ê ˆ dx w cf ( cos f( sin d cf ( cos f ( f( sin cos [f( ] sin ; y f( sin Ê f ( sin f( cos Ê f ( sin f( cos f ( sin f ( f( sin cos [f( ] cos. Therefore d d d ˆ dx dy w w Š cf( d cos sin b [f( ] cos sin b cf( d [f( ] r ˆ. d d d Thus, L dx ʈ dy Š d Ér ˆ d.. ( r v ( cos d sin c d (b r d c d v ˆ ˆ c c d (c r cos d sin v w d d w 5. r f(, Ÿ Ÿ Ê f ( Ê r ˆ w [f(] cf ( d Ê Length É[f( ] cf ( d d É[f( ] cf ( d d which is twice the length of the curve r f( for Ÿ Ÿ. w w. Agin r f( Ê r ˆ w [f(] cf ( d Ê Surfce Are [f( sin ] É[f( ] cf ( d d d [f( sin ] É[f( ] cf ( d d which is four times the re of the surfce generted by revolving w r f( bout the x-xis for Ÿ Ÿ. 7. x $ $ $ $ r cos d [( cos ] (cos d cos cos cos b(cos d r d [( cos ] d cos cos b d cos cos cos ˆ b sin b(cos ˆ b d (After considerble lgebr using cos cos ˆ b d the identity cos A cos A ˆ 5 8 cos cos cos sin cos d 5 8 sin sin sin sin ˆ cos cos d 8 sin sin $ ˆ 5 5 y $ r sin d $ [( cos ] (sin d ; u ( cos Ê du sin d ; Ê u ; r d Ê u d Ä u du $. Therefore the centroid is Bßyb ˆ 5 ß ; 8. r d d c d ; x ; $ r cos d $ cos d $ csin d r d $ r sin d $ sin d $ $ c cos d ˆ y. Therefore the centroid is xßyb ˆ ß. r d
8 .8 CONIC SECTIONS IN POLAR COORDINATES Section.8 Conic Sections in Polr Coordintes 7. r cos ˆ 5 r ˆ cos cos sin sin Ê 5 Ê r cos r sin 5 Ê x y 5 Ê x y Ê y x. r cos ˆ r ˆ cos cos sin sin Ê Ê r cos r sin Ê x y Ê x y Ê y x. r cos ˆ r ˆ cos cos sin sin Ê Ê r cos r sin Ê x y Ê x y Ê y x. r cos ˆ ˆ r cos ˆ r ˆ Ê Ê cos cos sin sin Ê r cos r sin Ê x y Ê x y 8 Ê y x 5. r cos ˆ r ˆ Ê cos cos sin sin Ê r cos r sin Ê x y Ê x y Ê y x. r cos ˆ r ˆ Ê cos cos sin sin Ê r cos r sin Ê x y Ê y x 7. r cos ˆ r ˆ Ê cos cos sin sin x y y x Ê r cos r sin Ê x y Ê Ê
9 8 Chpter Conic Sections nd Polr Coordintes 8. r cos ˆ r ˆ Ê cos cos sin sin x y y x Ê r cos r sin Ê x y Ê Ê 9. x y r cos r sin r Š cos sin r ˆ Ê Ê Ê cos cos sin sin Ê r cos ˆ. x y r cos r sin r Š cos sin r ˆ Ê Ê Ê cos cos sin sin Ê r cos ˆ. y 5 Ê r sin 5 Ê r sin 5 Ê r sin ( 5 Ê r cos ˆ ( 5 Ê r cos ˆ 5. x Ê r cos Ê r cos Ê r cos (. r ( cos 8 cos. r ( sin sin 5. r sin. r ˆ cos cos
10 Section.8 Conic Sections in Polr Coordintes 9. (x y Ê C (ß,. (x y Ê C ( ß, Ê r cos is the polr eqution Ê r cos is the polr eqution. x (y 5 5 Ê C (ß5, 5. x (y 7 9 Ê C (ß7, 7 Ê r sin is the polr eqution Ê r sin is the polr eqution 5. x x y Ê (x y. x x y Ê (x 8 y Ê C ( ß, Ê r cos is Ê C (8ß, 8 Ê r cos is the the polr eqution polr eqution 7. x y y Ê x ˆ y 8. x y y Ê x ˆ y 9 Ê C ˆ ß, Ê r sin is the Ê C ˆ ß, Ê r sin is the polr eqution polr eqution
11 7 Chpter Conic Sections nd Polr Coordintes ( ( cos cos 9. e, x Ê k Ê r ( ( sin sin. e, y Ê k Ê r (5 5 sin 5 sin. e 5, y Ê k Ê r ( 8 cos cos. e, x Ê k Ê r ˆ ( ˆ cos cos. e, x Ê k Ê r ˆ ( ˆ cos cos. e, x Ê k Ê r ˆ ( 5 ˆ 5 sin 5sin 5 5. e, x Ê k Ê r ˆ ( ˆ sin sin. e, y Ê k Ê r cos 7. r Ê e, k Ê x cos ˆ cos 8. r Ê e, k Ê x ; e b ke ˆ Ê Ê Ê Ê e 5 ˆ 5 ˆ 5 5 cos ˆ 5 cos ˆ cos e, k 5 x 5; e b ke ˆ e 9. r Ê r Ê Ê Ê Ê Ê Ê cos cos. r Ê r Ê e, k Ê x
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