Parametric Equations and Polar Coordinates

Transcription

1 Prmetric Equtions nd Polr Coordintes The Hle-Bopp comet, with its blue ion til nd white dust til, ppered in the sk in Mrch 997. In Section.6 ou will see how polr coordintes provide convenient eqution for the pth of this comet. Den Ketelsen So fr we hve described plne curves b giving s function of f or s function of t or b giving reltion between nd tht defines implicitl s function of f,. In this chpter we discuss two new methods for describing curves. Some curves, such s the ccloid, re best hndled when both nd re given in terms of third vrible t clled prmeter f t, tt. ther curves, such s the crdioid, hve their most convenient description when we use new coordinte sstem, clled the polr coordinte sstem. 635

2 636 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES. Curves Defined b Prmetric Equtions C (, )={ f(t), g(t)} Imgine tht prticle moves long the curve C shown in Figure. It is impossible to describe C b n eqution of the form f becuse C fils the Verticl Line Test. But the - nd -coordintes of the prticle re functions of time nd so we cn write f t nd tt. Such pir of equtions is often convenient w of describing curve nd gives rise to the following definition. Suppose tht nd re both given s functions of third vrible t (clled prmeter) b the equtions FIGURE f t tt (clled prmetric equtions). Ech vlue of t determines point,, which we cn plot in coordinte plne. As t vries, the point, f t, tt vries nd trces out curve C, which we cll prmetric curve. The prmeter t does not necessril represent time nd, in fct, we could use letter other thn t for the prmeter. But in mn pplictions of prmetric curves, t does denote time nd therefore we cn interpret, f t, tt s the position of prticle t time t. EXAMPLE Sketch nd identif the curve defined b the prmetric equtions t t t SLUTIN Ech vlue of t gives point on the curve, s shown in the tble. For instnce, if t, then, nd so the corresponding point is,. In Figure we plot the points, determined b severl vlues of the prmeter nd we join them to produce curve. t t= t= t= (, ) t=3 t=_ t=4 8 t=_ FIGURE This eqution in nd describes where the prticle hs been, but it doesn t tell us when the prticle ws t prticulr point. The prmetric equtions hve n dvntge the tell us when the prticle ws t point. The lso indicte the direction of the motion. A prticle whose position is given b the prmetric equtions moves long the curve in the direction of the rrows s t increses. Notice tht the consecutive points mrked on the curve pper t equl time intervls but not t equl distnces. Tht is becuse the prticle slows down nd then speeds up s t increses. It ppers from Figure tht the curve trced out b the prticle m be prb ol. This cn be confirmed b eliminting the prmeter t s follows. We obtin t from the second eqution nd substitute into the first eqution. This gives t t 4 3 nd so the curve represented b the given prmetric equtions is the prbol 4 3.

3 (8, 5) SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS 637 No restriction ws plced on the prmeter t in Emple, so we ssumed tht t could be n rel number. But sometimes we restrict t to lie in finite intervl. For instnce, the prmetric curve t t t t 4 FIGURE 3 (, ) shown in Figure 3 is the prt of the prbol in Emple tht strts t the point, nd ends t the point 8, 5. The rrowhed indictes the direction in which the curve is trced s t increses from to 4. In generl, the curve with prmetric equtions f t tt t b hs initil point f, t nd terminl point f b, tb. v EXAMPLE Wht curve is represented b the following prmetric equtions? π t= cos t sin t t t=π FIGURE 4 3π t= t (cos t, sin t) t= (, ) t=π SLUTIN If we plot points, it ppers tht the curve is circle. We cn confirm this impression b eliminting t. bserve tht cos t sin t Thus the point, moves on the unit circle. Notice tht in this emple the prmeter t cn be interpreted s the ngle (in rdins) shown in Figure 4. As t increses from to, the point, cos t, sin t moves once round the circle in the counterclockwise direction strting from the point,. EXAMPLE 3 Wht curve is represented b the given prmetric equtions? sin t cos t t t=, π, π (, ) SLUTIN Agin we hve sin t cos t so the prmetric equtions gin represent the unit circle. But s t increses from to, the point, sin t, cos t strts t, nd moves twice round the circle in the clockwise direction s indicted in Figure 5. FIGURE 5 Emples nd 3 show tht different sets of prmetric equtions cn represent the sme curve. Thus we distinguish between curve, which is set of points, nd prmetric curve, in which the points re trced in prticulr w. EXAMPLE 4 Find prmetric equtions for the circle with center h, k nd rdius r. SLUTIN If we tke the equtions of the unit circle in Emple nd multipl the epressions for nd b r, we get r cos t, r sin t. You cn verif tht these equtions represent circle with rdius r nd center the origin trced counterclockwise. We now shift h units in the -direction nd k units in the -direction nd obtin pr-

4 638 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES metric equtions of the circle (Figure 6) with center h, k nd rdius r: h r cos t k r sin t t r (h, k) FIGURE 6 =h+r cos t, =k+r sin t (_, ) (, ) v EXAMPLE 5 Sketch the curve with prmetric equtions sin t, sin t. FIGURE 7 SLUTIN bserve tht sin t nd so the point, moves on the prbol. But note lso tht, since sin t, we hve, so the prmetric equtions represent onl the prt of the prbol for which. Since sin t is periodic, the point, sin t, sin t moves bck nd forth infinitel often long the prbol from, to,. (See Figure 7.) TEC Module.A gives n ni m tion of the reltionship between motion long prmetric curve f t, tt nd motion long the grphs of f nd t s functions of t. Clicking on TRIG gives ou the fmil of prmetric curves cos bt c sin dt =cos t If ou choose b c d nd click on nimte, ou will see how the grphs of cos t nd sin t relte to the circle in Emple. If ou choose b c, d, ou will see grphs s in Figure 8. B clicking on nimte or moving the t-slider to the right, ou cn see from the color coding how motion long the grphs of cos t nd sin t corresponds to motion long the prmetric curve, which is clled Lissjous figure. t t FIGURE 8 =cos t =sin t =sin t Grphing Devices Most grphing clcultors nd computer grphing progrms cn be used to grph curves defined b prmetric equtions. In fct, it s instructive to wtch prmetric curve being drwn b grphing clcultor becuse the points re plotted in order s the corresponding prmeter vlues increse.

5 SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS _3 3 _3 FIGURE 9 EXAMPLE 6 Use grphing device to grph the curve 4 3. SLUTIN If we let the prmeter be t, then we hve the equtions t 4 3t t Using these prmetric equtions to grph the curve, we obtin Figure 9. It would be possible to solve the given eqution 4 3 for s four functions of nd grph them individull, but the prmetric equtions provide much esier method. In generl, if we need to grph n eqution of the form t, we cn use the prmetric equtions tt t Notice lso tht curves with equtions f (the ones we re most fmilir with grphs of functions) cn lso be regrded s curves with prmetric equtions t f t Grphing devices re prticulrl useful for sketching complicted curves. For instnce, the curves shown in Figures,, nd would be virtull impossible to produce b hnd..5.8 _ _.5 FIGURE =sin t+ cos 5t+ sin 3t 4 =cos t+ sin 5t+ cos 3t 4 _ FIGURE =sin t-sin.3t =cos t _.8 FIGURE =sin t+ sin 5t+ cos.3t 4 =cos t+ cos 5t+ sin.3t 4 ne of the most importnt uses of prmetric curves is in computer-ided design (CAD). In the Lbortor Project fter Section. we will investigte specil prmetric curves, clled Bézier curves, tht re used etensivel in mnufcturing, especill in the utomotive industr. These curves re lso emploed in specifing the shpes of letters nd other smbols in lser printers. The Ccloid TEC An nimtion in Module.B shows how the ccloid is formed s the circle moves. EXAMPLE 7 The curve trced out b point P on the circumference of circle s the circle rolls long stright line is clled ccloid (see Figure 3). If the circle hs rdius r nd rolls long the -is nd if one position of P is the origin, find prmetric equtions for the ccloid. P P FIGURE 3 P

6 64 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES P r r C(r, r) Q T SLUTIN We choose s prmeter the ngle of rottion of the circle when P is t the origin). Suppose the circle hs rotted through rdins. Becuse the circle hs been in contct with the line, we see from Figure 4 tht the distnce it hs rolled from the origin is rc PT r T Therefore the center of the circle is Cr, r. Let the coordintes of P be,. Then from Figure 4 we see tht T PQ r r sin r sin FIGURE 4 TC QC r r cos r cos Therefore prmetric equtions of the ccloid re r sin r cos A ne rch of the ccloid comes from one rottion of the circle nd so is described b. Although Equtions were derived from Figure 4, which illustrtes the cse where, it cn be seen tht these equtions re still vlid for other vlues of (see Eercise 39). Although it is possible to eliminte the prmeter from Equtions, the resulting Crtesin eqution in nd is ver complicted nd not s convenient to work with s the prmetric equtions. ccloid FIGURE 5 P P FIGURE 6 P P B P ne of the first people to stud the ccloid ws Glileo, who proposed tht bridges be built in the shpe of ccloids nd who tried to find the re under one rch of ccloid. Lter this curve rose in connection with the brchistochrone problem: Find the curve long which prticle will slide in the shortest time (under the influence of grvit) from point A to lower point B not directl beneth A. The Swiss mthemticin John Bernoulli, who posed this problem in 696, showed tht mong ll possible curves tht join A to B, s in Figure 5, the prticle will tke the lest time sliding from A to B if the curve is prt of n inverted rch of ccloid. The Dutch phsicist Hugens hd lred shown tht the ccloid is lso the solution to the tutochrone problem; tht is, no mtter where prticle P is plced on n inverted ccloid, it tkes the sme time to slide to the bottom (see Figure 6). Hugens proposed tht pendulum clocks (which he invented) should swing in ccloidl rcs becuse then the pendulum would tke the sme time to mke complete oscilltion whether it swings through wide or smll rc. v Fmilies of Prmetric Curves EXAMPLE 8 Investigte the fmil of curves with prmetric equtions cos t tn t sin t Wht do these curves hve in common? How does the shpe chnge s increses? SLUTIN We use grphing device to produce the grphs for the cses,,.5,.,,.5,, nd shown in Figure 7. Notice tht ll of these curves (ecept the cse ) hve two brnches, nd both brnches pproch the verticl smptote s pproches from the left or right.

7 SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS 64 =_ =_ =_.5 =_. = =.5 = = FIGURE 7 Members of the fmil =+cos t, = tn t+sin t, ll grphed in the viewing rectngle _4, 4 b _4, 4 When, both brnches re smooth; but when reches, the right brnch cquires shrp point, clled cusp. For between nd the cusp turns into loop, which becomes lrger s pproches. When, both brnches come together nd form circle (see Emple ). For between nd, the left brnch hs loop, which shrinks to become cusp when. For, the brnches become smooth gin, nd s increses further, the become less curved. Notice tht the curves with positive re reflections bout the -is of the corresponding curves with negtive. These curves re clled conchoids of Nicomedes fter the ncient Greek scholr Nicomedes. He clled them conchoids becuse the shpe of their outer brnches resembles tht of conch shell or mussel shell.. Eercises 4 Sketch the curve b using the prmetric equtions to plot points. Indicte with n rrow the direction in which the curve is trced s t increses.. t t, t t, t. t, t 3 4t, 3 t 3 3. cos t, sin t, t 4. e t t, e t t, t 5 () Sketch the curve b using the prmetric equtions to plot points. Indicte with n rrow the direction in which the curve is trced s t increses. (b) Eliminte the prmeter to find Crtesin eqution of the curve t, 3t 6. t, t, t 4 7. t, t, t 8. t, t 3, t 9. st,. t, t 3 8 () Eliminte the prmeter to find Crtesin eqution of the curve. (b) Sketch the curve nd indicte with n rrow the direction in which the curve is trced s the prmeter increses.. sin, cos,. cos, sin, 3. sin t, csc t, t 4. e t, 5. e t, t 6. st, 7. sinh t, t e t st cosh t 8. tn, sec, ; Grphing clcultor or computer required. Homework Hints vilble t stewrtclculus.com

8 64 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES 9 Describe the motion of prticle with position, s t vries in the given intervl cos t, sin t, t 3. sin t, 4 cos t, t 3. 5 sin t, cos t, t 5. sin t, cos t, t 5 7 Use the grphs of f t nd tt to sketch the prmetric curve f t, tt. Indicte with rrows the direction in which the curve is trced s t increses. 5. t t _ 3. Suppose curve is given b the prmetric equtions f t, tt, where the rnge of f is, 4 nd the rnge of t is, 3. Wht cn ou s bout the curve? Mtch the grphs of the prmetric equtions f t nd tt in () (d) with the prmetric curves lbeled I IV. Give resons for our choices. t t () I 7. t t t t (b) (c) t t II III 8. Mtch the prmetric equtions with the grphs lbeled I-VI. Give resons for our choices. (Do not use grphing device.) () t 4 t, t (b) t t, st (c) sin t, sint sin t (d) cos 5t, sin t (e) t sin 4t, t cos 3t sin t cos t (f), 4 t 4 t I II III t t (d) IV IV V VI t t

9 SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS 643 ; 9. Grph the curve sin. ; 3. Grph the curves 3 4 nd 3 4 nd find their points of intersection correct to one deciml plce. 3. () Show tht the prmetric equtions t where t, describe the line segment tht joins the points P, nd P,. (b) Find prmetric equtions to represent the line segment from, 7 to 3,. ; 3. Use grphing device nd the result of Eercise 3() to drw the tringle with vertices A,, B4,, nd C, Find prmetric equtions for the pth of prticle tht moves long the circle 4 in the mnner described. () nce round clockwise, strting t, (b) Three times round counterclockwise, strting t, (c) Hlfw round counterclockwise, strting t, 3 ; 34. () Find prmetric equtions for the ellipse b. [Hint: Modif the equtions of the circle in Emple.] (b) Use these prmetric equtions to grph the ellipse when 3 nd b,, 4, nd 8. (c) How does the shpe of the ellipse chnge s b vries? ; Use grphing clcultor or computer to reproduce the picture t 4 when P is t one of its lowest points, show tht prmetric equtions of the trochoid re r d sin r d cos Sketch the trochoid for the cses d r nd d r. 4. If nd b re fied numbers, find prmetric equtions for the curve tht consists of ll possible positions of the point P in the figure, using the ngle s the prmeter. Then eliminte the prm eter nd identif the curve. b P 4. If nd b re fied numbers, find prmetric equtions for the curve tht consists of ll possible positions of the point P in the figure, using the ngle s the prmeter. The line segment AB is tngent to the lrger circle. A b P B Compre the curves represented b the prmetric equtions. How do the differ? 37. () t 3, t (b) t 6, (c) e 3t, e t 38. () t, t (b) cos t, (c) e t, e t t 4 sec t 43. A curve, clled witch of Mri Agnesi, consists of ll possible positions of the point P in the figure. Show tht prmetric equtions for this curve cn be written s cot sin Sketch the curve. = C 39. Derive Equtions for the cse. 4. Let P be point t distnce d from the center of circle of rdius r. The curve trced out b P s the circle rolls long stright line is clled trochoid. (Think of the motion of point on spoke of biccle wheel.) The ccloid is the specil cse of trochoid with d r. Using the sme prmeter s for the ccloid nd, ssuming the line is the -is nd A P

10 644 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES 44. () Find prmetric equtions for the set of ll points P s shown in the figure such tht P AB. (This curve is clled the cissoid of Diocles fter the Greek scholr Diocles, who introduced the cissoid s grphicl method for constructing the edge of cube whose volume is twice tht of given cube.) (b) Use the geometric description of the curve to drw rough sketch of the curve b hnd. Check our work b using the prmetric equtions to grph the curve. P ; 45. Suppose tht the position of one prticle t time t is given b 3 sin t nd the position of second prticle is given b 3 cos t () Grph the pths of both prticles. How mn points of intersection re there? (b) Are n of these points of intersection collision points? In other words, re the prticles ever t the sme plce t the sme time? If so, find the collision points. (c) Describe wht hppens if the pth of the second prticle is given b 46. If projectile is fired with n initil velocit of v meters per second t n ngle bove the horizontl nd ir resistnce is ssumed to be negligible, then its position fter t seconds A cos t B sin t = t t 3 cos t sin t t is given b the prmetric equtions v cos t v sin t tt where t is the ccelertion due to grvit ( 9.8 ms ). () If gun is fired with 3 nd v 5 ms, when will the bullet hit the ground? How fr from the gun will it hit the ground? Wht is the mimum height reched b the bullet? ; (b) Use grphing device to check our nswers to prt (). Then grph the pth of the projectile for severl other vlues of the ngle to see where it hits the ground. Summrize our findings. (c) Show tht the pth is prbolic b eliminting the prmeter. ; 47. Investigte the fmil of curves defined b the prmetric equtions t, t 3 ct. How does the shpe chnge s c increses? Illustrte b grphing severl members of the fmil. ; 48. The swllowtil ctstrophe curves re defined b the prmetric equtions ct 4t 3, ct 3t 4. Grph severl of these curves. Wht fetures do the curves hve in common? How do the chnge when c increses? ; 49. Grph severl members of the fmil of curves with prmetric equtions t cos t, t sin t, where. How does the shpe chnge s increses? For wht vlues of does the curve hve loop? ; 5. Grph severl members of the fmil of curves sin t sin nt, cos t cos nt where n is positive integer. Wht fetures do the curves hve in common? Wht hppens s n increses? ; 5. The curves with equtions sin nt, b cos t re clled Lissjous figures. Investigte how these curves vr when, b, nd n vr. (Tke n to be positive integer.) ; 5. Investigte the fmil of curves defined b the prmetric equtions cos t, sin t sin ct, where c. Strt b letting c be positive integer nd see wht hppens to the shpe s c increses. Then eplore some of the possibilities tht occur when c is frction. LABRATRY PRJECT ; RUNNING CIRCLES ARUND CIRCLES In this project we investigte fmilies of curves, clled hpoccloids nd epiccloids, tht re generted b the motion of point on circle tht rolls inside or outside nother circle. C b P (, ) A. A hpoccloid is curve trced out b fied point P on circle C of rdius b s C rolls on the inside of circle with center nd rdius. Show tht if the initil position of P is, nd the prmeter is chosen s in the figure, then prmetric equtions of the hpoccloid re b cos b cos b b b sin b sin b b ; Grphing clcultor or computer required

11 SECTIN. CALCULUS WITH PARAMETRIC CURVES 645 TEC Look t Module.B to see how hpoccloids nd epi ccloids re formed b the motion of rolling circles.. Use grphing device (or the interctive grphic in TEC Module.B) to drw the grphs of hpoccloids with positive integer nd b. How does the vlue of ffect the grph? Show tht if we tke 4, then the prmetric equtions of the hpoccloid reduce to 4 cos 3 4 sin 3 This curve is clled hpoccloid of four cusps, or n stroid. 3. Now tr b nd nd, frction where n nd d hve no common fctor. First let n nd tr to determine grphicll the effect of the denomintor d on the shpe of the grph. Then let n vr while keeping d constnt. Wht hppens when n d? 4. Wht hppens if b nd is irrtionl? Eperiment with n irrtionl number like s or e. Tke lrger nd lrger vlues for nd speculte on wht would hppen if we were to grph the hpoccloid for ll rel vlues of. 5. If the circle C rolls on the outside of the fied circle, the curve trced out b P is clled n epiccloid. Find prmetric equtions for the epiccloid. 6. Investigte the possible shpes for epiccloids. Use methods similr to Problems 4.. Clculus with Prmetric Curves Hving seen how to represent curves b prmetric equtions, we now ppl the methods of clculus to these prmetric curves. In prticulr, we solve problems involving tngents, re, rc length, nd surfce re. Tngents Suppose f nd t re differentible functions nd we wnt to find the tngent line t point on the curve where is lso differentible function of. Then the Chin Rule gives If ddt, we cn solve for dd: d dt d d d dt If we think of the curve s being trced out b moving prticle, then ddt nd ddt re the verticl nd horizontl velocities of the prticle nd Formul ss tht the slope of the tngent is the rtio of these velocities. d d d dt d dt if d dt Note tht d d d dt d dt Eqution (which ou cn remember b thinking of cnceling the dt s) enbles us to find the slope dd of the tngent to prmetric curve without hving to eliminte the prmeter t. We see from tht the curve hs horizontl tngent when ddt (provided tht ddt ) nd it hs verticl tngent when ddt (provided tht ddt ). This informtion is useful for sketching prmetric curves. As we know from Chpter 4, it is lso useful to consider d d. This cn be found b replcing b dd in Eqution : d d dt d d d d d d d d dt

12 646 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES EXAMPLE A curve C is defined b the prmetric equtions t, t 3 3t. () Show tht C hs two tngents t the point (3, ) nd find their equtions. (b) Find the points on C where the tngent is horizontl or verticl. (c) Determine where the curve is concve upwrd or downwrd. (d) Sketch the curve. SLUTIN () Notice tht t 3 3t tt 3 when t or t s3. Therefore the point 3, on C rises from two vlues of the prmeter, t s3 nd t s3. This indictes tht C crosses itself t 3,. Since d ddt d ddt 3t 3 3 t t t the slope of the tngent when t s3 is dd 6(s3 ) s3, so the equtions of the tngents t 3, re s3 3 nd s3 3 t=_ (, ) t= (, _) FIGURE =œ 3(-3) (3, ) =_ œ 3(-3) (b) C hs horizontl tngent when dd, tht is, when ddt nd ddt. Since ddt 3t 3, this hppens when t, tht is, t. The corresponding points on C re, nd (, ). C hs verticl tngent when ddt t, tht is, t. (Note tht ddt there.) The corresponding point on C is (, ). (c) To determine concvit we clculte the second derivtive: d d d dt d d 3 t 3t d t 4t 3 dt Thus the curve is concve upwrd when t nd concve downwrd when t. (d) Using the informtion from prts (b) nd (c), we sketch C in Figure. v EXAMPLE () Find the tngent to the ccloid r sin, r cos t the point where 3. (See Emple 7 in Section..) (b) At wht points is the tngent horizontl? When is it verticl? SLUTIN () The slope of the tngent line is d d When 3, we hve dd dd r sin r cos sin cos r 3 sin 3 r 3 s3 r cos 3 r nd d d sin3 s3 cos3 s3

13 SECTIN. CALCULUS WITH PARAMETRIC CURVES 647 Therefore the slope of the tngent is s3 nd its eqution is r s3 r 3 The tngent is sketched in Figure. rs3 or s3 r s3 (_πr, r) (πr, r) (3πr, r) (5πr, r) π = 3 FIGURE πr 4πr (b) The tngent is horizontl when dd, which occurs when sin nd cos, tht is, n, n n integer. The corresponding point on the ccloid is n r, r. When n, both dd nd dd re. It ppers from the grph tht there re verticl tngents t those points. We cn verif this b using l Hospitl s Rule s follows: d lim ln d lim sin cos lim ln cos ln sin A similr computtion shows tht dd l s l n, so indeed there re verticl tngents when n, tht is, when nr. The limits of integrtion for t re found s usul with the Substitution Rule. When, t is either or. When b, t is the remining vlue. Ares We know tht the re under curve F from to b is A b F d, where F. If the curve is trced out once b the prmetric equtions f t nd tt, t, then we cn clculte n re formul b using the Sub stitution Rule for Definite Integrls s follows: A b d tt f t dt or tt f t dt v EXAMPLE 3 Find the re under one rch of the ccloid (See Figure 3.) r sin r cos πr SLUTIN ne rch of the ccloid is given b. Using the Substitution Rule with r cos nd d r cos d, we hve FIGURE 3 A r d r cos r cos d r cos d r cos cos d The result of Emple 3 ss tht the re under one rch of the ccloid is three times the re of the rolling circle tht genertes the ccloid (see Emple 7 in Section.). Glileo guessed this result but it ws first proved b the French mthemticin Robervl nd the Itlin mthemticin Torricelli. r [ cos cos ] d r [ 3 sin 4 sin ] r ( 3 ) 3r

14 648 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES Arc Length We lred know how to find the length L of curve C given in the form F, b. Formul 8..3 ss tht if F is continuous, then b L d d d Suppose tht C cn lso be described b the prmetric equtions f t nd tt, t, where ddt f t. This mens tht C is trversed once, from left to right, s t increses from to nd f, f b. Putting Formul into Formul nd using the Substitution Rule, we obtin b L d d d ddt d ddt dt dt Since ddt, we hve 3 L dt d dt d dt P P C P P i_ P i P n Even if Ccn t be epressed in the form F, Formul 3 is still vlid but we obtin it b polgonl pproimtions. We divide the prmeter intervl, into n subintervls of equl width t. If t, t, t,..., t n re the endpoints of these subintervls, then i f t i nd i tt i re the coordintes of points P i i, i tht lie on C nd the polgon with vertices P, P,..., P n pproimtes C. (See Figure 4.) As in Section 8., we define the length L of C to be the limit of the lengths of these pproimting polgons s n l : FIGURE 4 L lim nl n P ip i i The Men Vlue Theorem, when pplied to f on the intervl t i, t i, gives number t i * in t i, t i such tht f t i f t i f t i *t i t i If we let i i i nd i i i, this eqution becomes i f t i * t Similrl, when pplied to t, the Men Vlue Theorem gives number t i ** in t i, t i such tht i tt i ** t Therefore P ip i s i i s f t i *t tt i **t s f t i * tt i ** t nd so 4 L lim n l n s f t i * tt i ** t i

15 SECTIN. CALCULUS WITH PARAMETRIC CURVES 649 The sum in 4 resembles Riemnn sum for the function s f t tt but it is not ectl Riemnn sum becuse t i * t i ** in generl. Nevertheless, if f nd t re continuous, it cn be shown tht the limit in 4 is the sme s if t i * nd t i ** were equl, nmel, L s f t tt dt Thus, using Leibniz nottion, we hve the following result, which hs the sme form s Formul 3. 5 Theorem If curve C is described b the prmetric equtions f t, tt, t, where f nd t re continuous on, nd C is trversed ectl once s t increses from to, then the length of C is L dt dt d dt d Notice tht the formul in Theorem 5 is consistent with the generl formuls L ds nd ds d d of Section 8.. If we use the representtion of the unit circle given in Emple in Sec- EXAMPLE 4 tion., cos t sin t t then ddt sin t nd ddt cos t, so Theorem 5 gives L dt d dt d dt ssin t cos t dt dt s epected. If, on the other hnd, we use the representtion given in Emple 3 in Section., sin t cos t t then ddt cos t, ddt sin t, nd the integrl in Theorem 5 gives dt d dt d dt s4 cos t 4 sin t dt Notice tht the integrl gives twice the rc length of the circle becuse s t increses from to, the point sin t, cos t trverses the circle twice. In generl, when finding the length of curve C from prmetric representtion, we hve to be creful to ensure tht C is trversed onl once s t increses from to. v EXAMPLE 5 Find the length of one rch of the ccloid r sin, r cos. dt 4 SLUTIN From Emple 3 we see tht one rch is described b the prmeter intervl. Since d r cos d nd d r sin d

16 65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES we hve L d d d d d sr cos r sin d The result of Emple 5 ss tht the length of one rch of ccloid is eight times the rdius of the gener ting circle (see Figure 5). This ws first proved in 658 b Sir Christopher Wren, who lter becme the rchitect of St. Pul s Cthedrl in London. r L=8r sr cos cos sin d r s cos d To evlute this integrl we use the identit sin cos with, which gives cos sin. Since, we hve nd so sin. Therefore s cos s4 sin sin sin πr nd so L r sin d r cos] FIGURE 5 r 8r Surfce Are In the sme w s for rc length, we cn dpt Formul 8..5 to obtin formul for surfce re. If the curve given b the prmetric equtions f t, tt, t, is rotted bout the -is, where f, t re continuous nd tt, then the re of the resulting surfce is given b S d dt dt d 6 dt The generl smbolic formuls S dsnd S ds(formuls 8..7 nd 8..8) re still vlid, but for prmetric curves we use ds dt d dt d dt EXAMPLE 6 Show tht the surfce re of sphere of rdius r is 4r. SLUTIN The sphere is obtined b rotting the semicircle r cos t r sin t t bout the -is. Therefore, from Formul 6, we get S r sin t sr sin t r cos t dt r sin t sr sin t cos t dt r sin t rdt r cos t] 4r r sin tdt

17 SECTIN. CALCULUS WITH PARAMETRIC CURVES 65. Eercises Find dd.. t sin t, t t. t, st e t 3 6 Find n eqution of the tngent to the curve t the point corresponding to the given vlue of the prmeter. 3. 4t t, t 3 ; t 4. t t, t ; t 5. t cos t, t sin t; t 6. sin 3, cos 3 ; Find n eqution of the tngent to the curve t the given point b two methods: () without eliminting the prmeter nd (b) b first eliminting the prmeter. 7. ln t, t ;, 3 8. st, e t ;, e ; 9 Find n eqution of the tngent(s) to the curve t the given point. Then grph the curve nd the tngent(s) sin t, t t;,. cos t cos t, sin t sin t ;, 6 Find dd nd d d. For which vlues of t is the curve concve upwrd?. t, t t. t 3, t t 3. e t, te t 4. t, e t 5. sin t, 3 cos t, t 6. cos t, cos t, t 7 Find the points on the curve where the tngent is horizontl or verticl. If ou hve grphing device, grph the curve to check our work. 7. t 3 3t, t 3 8. t 3 3t, t 3 3t 9. cos, sin. e, cos 3 cos e ;. Use grph to estimte the coordintes of the rightmost point on the curve t t 6, e t. Then use clculus to find the ect coordintes. ;. Use grph to estimte the coordintes of the lowest point nd the leftmost point on the curve t 4 t, t t 4. Then find the ect coordintes. ; 3 4 Grph the curve in viewing rectngle tht displs ll the importnt spects of the curve. 3. t 4 t 3 t, 4. t 4 4t 3 8t, t 3 t t t 5. Show tht the curve cos t, sin t cos t hs two tngents t, nd find their equtions. Sketch the curve. ; 6. Grph the curve cos t cos t, sin t sin t to discover where it crosses itself. Then find equtions of both tngents t tht point. 7. () Find the slope of the tngent line to the trochoid r d sin, r d cos in terms of. (See Eercise 4 in Section..) (b) Show tht if d r, then the trochoid does not hve verticl tngent. 8. () Find the slope of the tngent to the stroid cos 3, sin 3 in terms of. (Astroids re eplored in the Lbortor Project on pge 644.) (b) At wht points is the tngent horizontl or verticl? (c) At wht points does the tngent hve slope or? 9. At wht points on the curve t 3, 4t t does the tngent line hve slope? 3. Find equtions of the tngents to the curve 3t, t 3 tht pss through the point 4, Use the prmetric equtions of n ellipse, cos, b sin,, to find the re tht it encloses. 3. Find the re enclosed b the curve t t, st nd the -is. 33. Find the re enclosed b the -is nd the curve e t, t t. 34. Find the re of the region enclosed b the stroid cos 3, sin 3. (Astroids re eplored in the Lbortor Project on pge 644.) 35. Find the re under one rch of the trochoid of Eercise 4 in Section. for the cse d r. ; Grphing clcultor or computer required CAS Computer lgebr sstem required. Homework Hints vilble t stewrtclculus.com

18 65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES 36. Let be the region enclosed b the loop of the curve in Emple. () Find the re of. (b) If is rotted bout the -is, find the volume of the resulting solid. (c) Find the centroid of. CAS where e is the eccentricit of the ellipse (e c, where c s b ). 54. Find the totl length of the stroid cos 3, sin 3, where. 55. () Grph the epitrochoid with equtions 37 4 Set up n integrl tht represents the length of the curve. Then use our clcultor to find the length correct to four deciml plces. 37. t e t, t e t, t 38. t t, t 4, t t sin t, cos t, t 4 4. t st, t st, t 4 44 Find the ect length of the curve. 4. 3t, 4 t 3, t 4. e t e t, 5 t, t t sin t, t cos t, t cos t cos 3t, 3 sin t sin 3t, ; Grph the curve nd find its length. 45. e t cos t, e t sin t, t 46. cos t ln(tn t), sin t, 4 t 34 ; 47. Grph the curve sin t sin.5t, cos t nd find its length correct to four deciml plces. 48. Find the length of the loop of the curve 3t t 3, 3t. 49. Use Simpson s Rule with n 6 to estimte the length of the curve t e t, t e t, 6 t In Eercise 43 in Section. ou were sked to derive the prmetric equtions cot, sin for the curve clled the witch of Mri Agnesi. Use Simpson s Rule with n 4 to estimte the length of the rc of this curve given b Find the distnce trveled b prticle with position, s t vries in the given time intervl. Compre with the length of the curve. 5. sin t, cos t, 5. cos t, cos t, t 3 t 4 t 53. Show tht the totl length of the ellipse sin, b cos, b, is L 4 s e sin d CAS Wht prmeter intervl gives the complete curve? (b) Use our CAS to find the pproimte length of this curve. 56. A curve clled Cornu s spirl is defined b the prmetric equtions where C nd S re the Fresnel functions tht were intro duced in Chpter 5. () Grph this curve. Wht hppens s t l nd s t l? (b) Find the length of Cornu s spirl from the origin to the point with prmeter vlue t Set up n integrl tht represents the re of the surfce obtined b rotting the given curve bout the -is. Then use our clcultor to find the surfce re correct to four deciml plces. 57. t sin t, t cos t, 58. sin t, sin t, 59. te t, t e t, t 6. t t 3, t t 4, t 6 63 Find the ect re of the surfce obtined b rotting the given curve bout the -is. 6. t 3, t, t 6. 3t t 3, 3t, t 63. cos 3, sin 3, ; 64. Grph the curve If this curve is rotted bout the -is, find the re of the resulting surfce. (Use our grph to help find the correct prmeter intervl.) Find the surfce re generted b rotting the given curve bout the -is t, t 3, cos t 4cost sin t 4sint Ct t cosu du St t sinu du cos cos t t t 5 sin sin

19 LABRATRY PRJECT BÉZIER CURVES e t t, 4e t, t 67. If f is continuous nd f t for t b, show tht the prmetric curve f t, tt, t b, cn be put in the form F. [Hint: Show tht f eists.] 68. Use Formul to derive Formul 7 from Formul 8..5 for the cse in which the curve cn be represented in the form F, b. 69. The curvture t point P of curve is defined s d ds where is the ngle of inclintion of the tngent line t P, s shown in the figure. Thus the curvture is the bsolute vlue of the rte of chnge of with respect to rc length. It cn be regrded s mesure of the rte of chnge of direction of the curve t P nd will be studied in greter detil in Chpter 3. () For prmetric curve t, t, derive the formul 3 where the dots indicte derivtives with respect to t, so ddt. [Hint: Use tn dd nd Formul to find ddt. Then use the Chin Rule to find dds.] (b) B regrding curve f s the prmetric curve, f, with prmeter, show tht the formul in prt () becomes d d dd 3 7. () Use the formul in Eercise 69(b) to find the curvture of the prbol t the point,. (b) At wht point does this prbol hve mimum curvture? 7. Use the formul in Eercise 69() to find the curvture of the ccloid sin, cos t the top of one of its rches. 7. () Show tht the curvture t ech point of stright line is. (b) Show tht the curvture t ech point of circle of rdius r is r. 73. A string is wound round circle nd then unwound while being held tut. The curve trced b the point P t the end of the string is clled the involute of the circle. If the circle hs rdius r nd center nd the initil position of P is r,, nd if the prmeter is chosen s in the figure, show tht prmetric equtions of the involute re rcos sin r rsin cos 74. A cow is tied to silo with rdius r b rope just long enough to rech the opposite side of the silo. Find the re vilble for grzing b the cow. T P P LABRATRY PRJECT ; BÉZIER CURVES Bézier curves re used in computer-ided design nd re nmed fter the French mthemticin Pierre Bézier (9 999), who worked in the utomotive industr. A cubic Bézier curve is determined b four control points, P,, P,, P,, nd P 3 3, 3, nd is defined b the prmetric equtions t 3 3 t t 3 t t 3 t 3 t 3 3 t t 3 t t 3 t 3 ; Grphing clcultor or computer required

20 654 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES where t. Notice tht when t we hve,, nd when t we hve, 3, 3, so the curve strts t P nd ends t P 3.. Grph the Bézier curve with control points P 4,, P 8, 48, P 5, 4, nd P 3 4, 5. Then, on the sme screen, grph the line segments P P, P P, nd P P 3. (Eercise 3 in Section. shows how to do this.) Notice tht the middle control points P nd P don t lie on the curve; the curve strts t P, heds towrd P nd P without reching them, nd ends t. P 3. From the grph in Problem, it ppers tht the tngent t P psses through P nd the tngent t P 3 psses through P. Prove it. 3. Tr to produce Bézier curve with loop b chnging the second control point in Problem. 4. Some lser printers use Bézier curves to represent letters nd other smbols. Eperiment with control points until ou find Bézier curve tht gives resonble representtion of the letter C. 5. More complicted shpes cn be represented b piecing together two or more Bézier curves. Suppose the first Bézier curve hs control points P, P, P, P 3 nd the second one hs control points P 3, P 4, P 5, P 6. If we wnt these two pieces to join together smoothl, then the tngents t P 3 should mtch nd so the points P, P 3, nd P 4 ll hve to lie on this common tngent line. Using this principle, find control points for pir of Bézier curves tht represent the letter S..3 Polr Coordintes r polr is FIGURE +π (_r, ) P(r, ) (r, ) A coordinte sstem represents point in the plne b n ordered pir of numbers clled coordintes. Usull we use Crtesin coordintes, which re directed distnces from two perpendiculr es. Here we describe coordinte sstem introduced b Newton, clled the polr coordinte sstem, which is more convenient for mn purposes. We choose point in the plne tht is clled the pole (or origin) nd is lbeled. Then we drw r (hlf-line) strting t clled the polr is. This is is usull drwn horizontll to the right nd corresponds to the positive -is in Crtesin coordintes. If Pis n other point in the plne, let rbe the distnce from to Pnd let be the ngle (usull mesured in rdins) between the polr is nd the line P s in Figure. Then the point P is represented b the ordered pir r, nd r, re clled polr coordintes of P. We use the convention tht n ngle is positive if mesured in the counterclockwise direction from the polr is nd negtive in the clockwise direction. If P, then r nd we gree tht, represents the pole for n vlue of. We etend the mening of polr coordintes r, to the cse in which r is negtive b greeing tht, s in Figure, the points r, nd r, lie on the sme line through nd t the sme distnce r from, but on opposite sides of. If r, the point r, lies in the sme qudrnt s ; if r, it lies in the qudrnt on the opposite side of the pole. Notice tht r, represents the sme point s r,. FIGURE EXAMPLE Plot the points whose polr coordintes re given. (), 54 (b), 3 (c), 3 (d) 3, 34

21 SECTIN.3 PLAR CRDINATES 655 SLUTIN The points re plotted in Figure 3. In prt (d) the point 3, 34 is locted three units from the pole in the fourth qudrnt becuse the ngle 34 is in the second qudrnt nd r 3 is negtive. 5π 4 5π, 4 (, 3π) 3π π _ 3 3π 4 FIGURE 3 π, _ 3 3π _3, 4 In the Crtesin coordinte sstem ever point hs onl one representtion, but in the polr coordinte sstem ech point hs mn representtions. For instnce, the point, 54 in Emple () could be written s, 34or, 34or, 4. (See Figure 4.) 5π 4 _ 3π 4 3π 4 π 4 5π, 4 3π, _ 4 3π, 4 π _, 4 FIGURE 4 In fct, since complete counterclockwise rottion is given b n ngle, the point represented b polr coordintes r, is lso represented b r, n nd r, n r P(r, )=P(, ) where n is n integer. The connection between polr nd Crtesin coordintes cn be seen from Figure 5, in which the pole corresponds to the origin nd the polr is coincides with the positive -is. If the point Phs Crtesin coordintes, nd polr coordintes r,, then, from the figure, we hve nd so cos r sin r FIGURE 5 r cos r sin Although Equtions were deduced from Figure 5, which illustrtes the cse where r nd, these equtions re vlid for ll vlues of r nd. (See the generl definition of sin nd cos in Appendi D.) Equtions llow us to find the Crtesin coordintes of point when the polr coordintes re known. To find r nd when nd re known, we use the equtions

22 656 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES r tn which cn be deduced from Equtions or simpl red from Figure 5. EXAMPLE Convert the point, 3 from polr to Crtesin coordintes. SLUTIN Since r nd 3, Equtions give r cos cos 3 r sin sin s3 3 Therefore the point is (, s3 ) in Crtesin coordintes. s3 EXAMPLE 3 Represent the point with Crtesin coordintes, in terms of polr coordintes. SLUTIN If we choose r to be positive, then Equtions give r s s s tn Since the point, lies in the fourth qudrnt, we cn choose 4 or 74. Thus one possible nswer is (s, 4) ; nother is s, 74. NTE Equtions do not uniquel determine when nd re given becuse, s increses through the intervl, ech vlue of tn occurs twice. Therefore, in converting from Crtesin to polr coordintes, it s not good enough just to find r nd tht stisf Equtions. As in Emple 3, we must choose so tht the point r, lies in the correct qudrnt. r= r= r= r=4 Polr Curves The grph of polr eqution r f, or more generll Fr,, consists of ll points P tht hve t lest one polr representtion r, whose coordintes stisf the eqution. FIGURE 6 v EXAMPLE 4 Wht curve is represented b the polr eqution r? SLUTIN The curve consists of ll points r, with r. Since r represents the distnce from the point to the pole, the curve r represents the circle with center nd rdius. In generl, the eqution r represents circle with center nd rdius. (See Figure 6.)

23 SECTIN.3 PLAR CRDINATES 657 (3, ) EXAMPLE 5 Sketch the polr curve. = (_, ) (_, ) FIGURE 7 (, ) (, ) SLUTIN This curve consists of ll points r, such tht the polr ngle is rdin. It is the stright line tht psses through nd mkes n ngle of rdin with the polr is (see Figure 7). Notice tht the points r, on the line with r re in the first qudrnt, wheres those with r re in the third qudrnt. EXAMPLE 6 () Sketch the curve with polr eqution r cos. (b) Find Crtesin eqution for this curve. SLUTIN () In Figure 8 we find the vlues of r for some convenient vlues of nd plot the corresponding points r,. Then we join these points to sketch the curve, which ppers to be circle. We hve used onl vlues of between nd, since if we let increse beond, we obtin the sme points gin. FIGURE 8 Tble of vlues nd grph of r= cos 6 s3 s r cos s s3 π, π, 3 π _, 3 π œ, 4 π œ, 3 6 (, ) 5π _ œ, 3 3π 6 _ œ, 4 (b) To convert the given eqution to Crtesin eqution we use Equtions nd. From r cos we hve cos r, so the eqution r cos becomes r r, which gives r Completing the squre, we obtin or which is n eqution of circle with center, nd rdius. P Figure 9 shows geometricl illustrtion tht the circle in Emple 6 hs the eqution r cos. The ngle PQ is right ngle (Wh?) nd so r cos. r Q FIGURE 9

24 658 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES r π π 3π π FIGURE r=+sin in Crtesin coordintes, π v EXAMPLE 7 Sketch the curve r sin. SLUTIN Insted of plotting points s in Emple 6, we first sketch the grph of r sin in Crtesin coordintes in Figure b shifting the sine curve up one unit. This enbles us to red t glnce the vlues of r tht correspond to incresing vlues of. For instnce, we see tht s increses from to, r (the distnce from ) increses from to, so we sketch the corresponding prt of the polr curve in Figure (). As increses from to, Figure shows tht r decreses from to, so we sketch the net prt of the curve s in Figure (b). As increses from to 3, r decreses from to s shown in prt (c). Finll, s increses from 3 to, r increses from to s shown in prt (d). If we let increse beond or decrese beond, we would simpl re trce our pth. Putting together the prts of the curve from Figure () (d), we sketch the complete curve in prt (e). It is clled crdioid becuse it s shped like hert. = π = π = =π =π =π = 3π = 3π () (b) (c) (d) (e) FIGURE Stges in sketching the crdioid r=+sin TEC Module.3 helps ou see how polr curves re trced out b showing nimtions similr to Figures 3. EXAMPLE 8 Sketch the curve r cos. SLUTIN As in Emple 7, we first sketch r cos,, in Crtesin coordintes in Figure. As increses from to 4, Figure shows tht r decreses from to nd so we drw the corresponding portion of the polr curve in Figure 3 (indicted b!). As increses from 4 to, r goes from to. This mens tht the distnce from increses from to, but insted of being in the first qudrnt this portion of the polr curve (indicted lies on the opposite side of the pole in the third qudrnt. The reminder of the curve is drwn in similr fshion, with the rrows nd numbers indicting the order in which the portions re trced out. The resulting curve hs four loops nd is clled four-leved rose. r = π! $ % * = 3π 4 $ & ^! = π 4 π 4 π 3π 4 # ^ & 5π 4 3π 7π 4 π =π # 8 = FIGURE r=cos in Crtesin coordintes FIGURE 3 Four-leved rose r=cos

25 Smmetr SECTIN.3 PLAR CRDINATES 659 When we sketch polr curves it is sometimes helpful to tke dvntge of smmetr. The following three rules re eplined b Figure 4. () If polr eqution is unchnged when is replced b, the curve is sm metric bout the polr is. (b) If the eqution is unchnged when r is replced b r, or when is replced b, the curve is smmetric bout the pole. (This mens tht the curve remins unchnged if we rotte it through 8 bout the origin.) (c) If the eqution is unchnged when is replced b, the curve is sm metric bout the verticl line. (r, ) (r, π- ) (r, ) _ (_r, ) (r, ) π- (r, _ ) () (b) (c) FIGURE 4 The curves sketched in Emples 6 nd 8 re smmetric bout the polr is, since cos cos. The curves in Emples 7 nd 8 re smmetric bout becuse sin sin nd cos cos. The four-leved rose is lso smmetric bout the pole. These smmetr properties could hve been used in sketching the curves. For instnce, in Emple 6 we need onl hve plotted points for nd then reflected bout the polr is to obtin the complete circle. Tngents to Polr Curves To find tngent line to polr curve r f, we regrd s prmeter nd write its prmetric equtions s r cos f cos r sin f sin Then, using the method for finding slopes of prmetric curves (Eqution..) nd the Product Rule, we hve 3 d d d d d d dr sin r cos d dr cos r sin d We locte horizontl tngents b finding the points where dd (provided tht dd ). Likewise, we locte verticl tngents t the points where dd (provided tht dd ). Notice tht if we re looking for tngent lines t the pole, then r nd Eqution 3 simplifies to d dr tn d if d

26 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES For instnce, in Emple 8 we found tht r cos when 4 or 34. This mens tht the lines 4 nd 34 (or nd ) re tngent lines to r cos t the origin. EXAMPLE 9 () For the crdioid r sin of Emple 7, find the slope of the tngent line when 3. (b) Find the points on the crdioid where the tngent line is horizontl or verticl. SLUTIN Using Eqution 3 with r sin, we hve d d dr sin r cos d dr cos r sin d cos sin sin cos cos cos sin sin cos sin cos sin sin sin sin sin () The slope of the tngent t the point where 3 is d cos3 sin3 d 3 sin3 sin3 ( s3 ) ( s3)( s3 ) s3 s3 ( s3 )( s3 ) s3 (b) bserve tht d cos sin d when, 3, 7 6, 6 d d sin sin when 3, 6, 5 6 π, m=_ œ 3 π +, 3, (, 76) (, 6) Therefore there re horizontl tngents t the points,, nd verticl tngents t ( 3, 6) nd ( 3, 56). When 3, both dd nd dd re, so we must be creful. Using l Hospitl s Rule, we hve d lim l3 d sin cos lim lim l3 sin l3 sin 3 5π, 6 (, ) 3 π, 6 3 lim l3 cos sin 3 sin lim l3 cos 7π π,, 6 6 FIGURE 5 Tngent lines for r=+sin B smmetr, d lim l3 d Thus there is verticl tngent line t the pole (see Figure 5).

27 SECTIN.3 PLAR CRDINATES 66 NTE Insted of hving to remember Eqution 3, we could emplo the method used to derive it. For instnce, in Emple 9 we could hve written Then we hve d d r cos sin cos cos sin r sin sin sin sin sin dd dd cos sin cos sin cos which is equivlent to our previous epression. Grphing Polr Curves with Grphing Devices cos sin sin cos Although it s useful to be ble to sketch simple polr curves b hnd, we need to use grphing clcultor or computer when we re fced with curve s complicted s the ones shown in Figures 6 nd _ FIGURE 6 r=sin@(.4 )+cos$(.4 ) _.7 FIGURE 7 r=sin@(. )+cos#(6 ) Some grphing devices hve commnds tht enble us to grph polr curves directl. With other mchines we need to convert to prmetric equtions first. In this cse we tke the polr eqution r f nd write its prmetric equtions s r cos f cos Some mchines require tht the prmeter be clled t rther thn. EXAMPLE Grph the curve r sin85. r sin f sin SLUTIN Let s ssume tht our grphing device doesn t hve built-in polr grphing commnd. In this cse we need to work with the corresponding prmetric equtions, which re r cos sin85 cos r sin sin85 sin In n cse we need to determine the domin for. So we sk ourselves: How mn complete rottions re required until the curve strts to repet itself? If the nswer is n, then sin 8 n 5 sin 8 5 6n 5 sin 8 5 nd so we require tht 6n5 be n even multiple of. This will first occur when n 5. Therefore we will grph the entire curve if we specif tht.

28 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES _ FIGURE 8 r=sin(8 /5) _ In Eercise 53 ou re sked to prove nlticll wht we hve discovered from the grphs in Figure 9. Switching from to t, we hve the equtions sin8t5 cos t sin8t5 sin t t nd Figure 8 shows the resulting curve. Notice tht this rose hs 6 loops. v EXAMPLE Investigte the fmil of polr curves given b r c sin. How does the shpe chnge s c chnges? (These curves re clled limçons, fter French word for snil, becuse of the shpe of the curves for certin vlues of c.) SLUTIN Figure 9 shows computer-drwn grphs for vrious vlues of c. For c there is loop tht decreses in size s c decreses. When c the loop disppers nd the curve becomes the crdioid tht we sketched in Emple 7. For c between nd the crdioid s cusp is smoothed out nd becomes dimple. When c de creses from to, the limçon is shped like n ovl. This ovl becomes more circulr s c l, nd when c the curve is just the circle r. c=.7 c= c=.7 c=.5 c=. c=.5 c=_ c= c=_. c=_.5 c=_.8 c=_ FIGURE 9 Members of the fmil of limçons r=+c sin The remining prts of Figure 9 show tht s c becomes negtive, the shpes chnge in reverse order. In fct, these curves re reflections bout the horizontl is of the corresponding curves with positive c. Limçons rise in the stud of plnetr motion. In prticulr, the trjector of Mrs, s viewed from the plnet Erth, hs been modeled b limçon with loop, s in the prts of Figure 9 with. c.3 Eercises Plot the point whose polr coordintes re given. Then find two other pirs of polr coordintes of this point, one with r nd one with r.. (), 3 (b), 34 (c),. (), 74 (b) 3, 6 (c), 3 4 Plot the point whose polr coordintes re given. Then find the Crtesin coordintes of the point. 3. (), (b) (, 3) (c), () (s, 54) (b), 5 (c), The Crtesin coordintes of point re given. (i) Find polr coordintes r, of the point, where r nd. (ii) Find polr coordintes r, of the point, where r nd. 5. (), (b) (, s3 ) 6. () (3s3, 3) (b), ; Grphing clcultor or computer required. Homework Hints vilble t stewrtclculus.com

29 SECTIN.3 PLAR CRDINATES Sketch the region in the plne consisting of points whose polr coordintes stisf the given conditions r r, 3 9. r, r 3, r 3, r, 3. Find the distnce between the points with polr coordintes, 3 nd 4, Find formul for the distnce between the points with polr coordintes r, nd r,. 5 Identif the curve b finding Crtesin eqution for the curve. 5. r 5 6. r 4 sec 7. r cos r cos. r tn sec 6 Find polr eqution for the curve represented b the given Crtesin eqution c For ech of the described curves, decide if the curve would be more esil given b polr eqution or Crtesin eqution. Then write n eqution for the curve. 7. () A line through the origin tht mkes n ngle of 6 with the positive -is (b) A verticl line through the point 3, 3 8. () A circle with rdius 5 nd center, 3 (b) A circle centered t the origin with rdius 4 4. r 9 sin 4. r cos r sin r 45. r cos 46. r 3 4 cos The figure shows grph of r s function of in Crtesin coordintes. Use it to sketch the corresponding polr curve r r π π 49. Show tht the polr curve r 4 sec (clled conchoid) hs the line s verticl smptote b showing tht lim r l. Use this fct to help sketch the conchoid. 5. Show tht the curve r csc (lso conchoid) hs the line s horizontl smptote b showing tht lim r l. Use this fct to help sketch the conchoid. 5. Show tht the curve r sin tn (clled cissoid of Diocles) hs the line s verticl smptote. Show lso tht the curve lies entirel within the verticl strip. Use these fcts to help sketch the cissoid. 5. Sketch the curve 3 4. π π 53. () In Emple the grphs suggest tht the limçon r c sin hs n inner loop when c. Prove tht this is true, nd find the vlues of tht correspond to the inner loop. (b) From Figure 9 it ppers tht the limçon loses its dimple when c. Prove this. 54. Mtch the polr equtions with the grphs lbeled I VI. Give resons for our choices. (Don t use grphing device.) () r s, 6 (b) r, 6 (c) r cos3 (d) r cos (e) r sin 3 (f) r sin 3 _ I II III 9 46 Sketch the curve with the given polr eqution b first sketching the grph of r s function of in Crtesion coordintes. 9. r sin 3. r cos 3. r cos 3. r cos 33. r, 34. r ln, 35. r 4 sin r cos r cos r 3 cos r sin 4. r sin IV V VI

30 664 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES 55 6 Find the slope of the tngent line to the given polr curve t the point specified b the vlue of. 55. r sin, r sin, 57. r, 58. r cos3, 59. r cos, 4 6. r cos, Find the points on the given curve where the tngent line is horizontl or verticl. 6. r 3 cos 6. r sin 63. r cos 64. r e 65. Show tht the polr eqution r sin b cos, where b, represents circle, nd find its center nd rdius. 66. Show tht the curves r sin nd r cos intersect t right ngles. ; 67 7 Use grphing device to grph the polr curve. Choose the prmeter intervl to mke sure tht ou produce the entire curve. 67. r sin (nephroid of Freeth) 68. r s.8 sin (hippopede) 69. r e sin cos4 (butterfl curve) 7. (vlentine curve) 7. r cos 999 (PcMn curve) 7. r tn cot r sin 4 cos4 3 ; 73. How re the grphs of r sin 6 nd r sin 3 relted to the grph of r sin? In generl, how is the grph of r f relted to the grph of r f? ; 74. Use grph to estimte the -coordinte of the highest points on the curve r sin. Then use clculus to find the ect vlue. ; 75. Investigte the fmil of curves with polr equtions r c cos, where c is rel number. How does the shpe chnge s c chnges? ; 76. Investigte the fmil of polr curves where n is positive integer. How does the shpe chnge s n increses? Wht hppens s n becomes lrge? Eplin the shpe for lrge n b considering the grph of r s function of in Crtesin coordintes. 77. Let P be n point (ecept the origin) on the curve r f. If is the ngle between the tngent line t P nd the rdil line P, show tht tn r drd [Hint: bserve tht in the figure.] r cos n r=f( ) 78. () Use Eercise 77 to show tht the ngle between the tngent line nd the rdil line is 4 t ever point on the curve r e. ; (b) Illustrte prt () b grphing the curve nd the tngent lines t the points where nd. (c) Prove tht n polr curve r f with the propert tht the ngle between the rdil line nd the tngent line is constnt must be of the form r Ce k, where C nd k re constnts. P ÿ LABRATRY PRJECT ; FAMILIES F PLAR CURVES In this project ou will discover the interesting nd beutiful shpes tht members of fmilies of polr curves cn tke. You will lso see how the shpe of the curve chnges when ou vr the constnts.. () Investigte the fmil of curves defined b the polr equtions r sin n, where n is positive integer. How is the number of loops relted to n? (b) Wht hppens if the eqution in prt () is replced b? r sin n. A fmil of curves is given b the equtions r c sin n, where c is rel number nd n is positive integer. How does the grph chnge s n increses? How does it chnge s c chnges? Illustrte b grphing enough members of the fmil to support our conclusions. ; Grphing clcultor or computer required

31 SECTIN.4 AREAS AND LENGTHS IN PLAR CRDINATES A fmil of curves hs polr equtions r cos cos Investigte how the grph chnges s the number chnges. In prticulr, ou should identif the trnsitionl vlues of for which the bsic shpe of the curve chnges. 4. The stronomer Giovnni Cssini (65 7) studied the fmil of curves with polr equtions r 4 c r cos c 4 4 where nd c re positive rel numbers. These curves re clled the ovls of Cssini even though the re ovl shped onl for certin vlues of nd c. (Cssini thought tht these curves might represent plnetr orbits better thn Kepler s ellipses.) Investigte the vriet of shpes tht these curves m hve. In prticulr, how re nd c relted to ech other when the curve splits into two prts?.4 Ares nd Lengths in Polr Coordintes r In this section we develop the formul for the re of region whose boundr is given b polr eqution. We need to use the formul for the re of sector of circle: A r FIGURE =b b r=f( ) = where, s in Figure, r is the rdius nd is the rdin mesure of the centrl ngle. Formul follows from the fct tht the re of sector is proportionl to its centrl ngle: A r r. (See lso Eercise 35 in Section 7.3.) Let be the region, illustrted in Figure, bounded b the polr curve r f nd b the rs nd b, where f is positive continuous function nd where b. We divide the intervl, b into subintervls with endpoints,,,..., n nd equl width. The rs i then divide into n smller regions with centrl ngle i i. If we choose i * in the ith subintervl i, i, then the re A i of the ith region is pproimted b the re of the sector of circle with centrl ngle nd rdius f i *. (See Figure 3.) Thus from Formul we hve FIGURE A i f i * f( i*) = i = i- nd so n pproimtion to the totl re A of is =b Î FIGURE 3 = A n i f i * It ppers from Figure 3 tht the pproimtion in improves s n l. But the sums in re Riemnn sums for the function t f, so lim n l n i f i * b f d

32 666 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES It therefore ppers plusible (nd cn in fct be proved) tht the formul for the re A of the polr region is 3 A b f d Formul 3 is often written s 4 A b r d with the understnding tht r f. Note the similrit between Formuls nd 4. When we ppl Formul 3 or 4 it is helpful to think of the re s being swept out b rotting r through tht strts with ngle nd ends with ngle b. r=cos = π 4 v EXAMPLE Find the re enclosed b one loop of the four-leved rose r cos. SLUTIN The curve r cos ws sketched in Emple 8 in Section.3. Notice from Figure 4 tht the region enclosed b the right loop is swept out b r tht rottes from 4 to 4. Therefore Formul 4 gives FIGURE 4 = 5π 6 =_ π 4 r=3 sin = π 6 A 4 4 A 4 r d 4 4 cos d 4 cos d cos 4 d [ 4 4 sin 4] v EXAMPLE Find the re of the region tht lies inside the circle r 3sin nd outside the crdioid r sin. SLUTIN The crdioid (see Emple 7 in Section.3) nd the circle re sketched in Figure 5 nd the desired region is shded. The vlues of nd b in Formul 4 re determined b finding the points of intersection of the two curves. The intersect when 3sin sin, which gives sin, so 6, 56. The desired re cn be found b subtrcting the re inside the crdioid between 6 nd 56 from the re inside the circle from 6 to 56. Thus 8 r=+sin A 56 3sin d 56 sin d 6 6 FIGURE 5 Since the region is smmetric bout the verticl is, we cn write A 6 9sin d 6 8 sin sin d 6 sin sin d 3 4 cos sin d sin 6 [becuse cos ] 3 sin cos ] 6

33 =b r=g( ) = FIGURE 6 r=f( ) SECTIN.4 AREAS AND LENGTHS IN PLAR CRDINATES 667 Emple illustrtes the procedure for finding the re of the region bounded b two polr curves. In generl, let be region, s illustrted in Figure 6, tht is bounded b curves with polr equtions r f, r t,, nd b, where f t nd b. The re A of is found b subtrcting the re inside r t from the re inside r f, so using Formul 3 we hve A b f d b t d b ( f t ) d r= FIGURE 7 π, 3 π, 6 r=cos CAUTIN The fct tht single point hs mn representtions in polr coordintes sometimes mkes it difficult to find ll the points of intersection of two polr curves. For instnce, it is obvious from Figure 5 tht the circle nd the crdioid hve three points of intersection; however, in Emple we solved the equtions r 3sin nd r sin nd found onl two such points, ( 3, 6) nd ( 3, 56). The origin is lso point of intersection, but we cn t find it b solving the equtions of the curves becuse the origin hs no single representtion in polr coordintes tht stisfies both equtions. Notice tht, when represented s, or,, the origin stisfies r 3sin nd so it lies on the circle; when represented s, 3, it stisfies r sin nd so it lies on the crdioid. Think of two points moving long the curves s the prmeter vlue increses from to. n one curve the origin is reched t nd ; on the other curve it is reched t 3. The points don t collide t the origin becuse the rech the origin t different times, but the curves intersect there nonetheless. Thus, to find ll points of intersection of two polr curves, it is recommended tht ou drw the grphs of both curves. It is especill convenient to use grphing clcultor or computer to help with this tsk. EXAMPLE 3 Find ll points of intersection of the curves r cos nd r. SLUTIN If we solve the equtions r cos nd r, we get cos nd, therefore, 3, 53, 73, 3. Thus the vlues of between nd tht stisf both equtions re 6, 56, 76, 6. We hve found four points of inter - section: (, 6), (, 56), (, 76), nd (, 6). However, ou cn see from Figure 7 tht the curves hve four other points of intersection nmel, (, 3), (, 3), (, 43), nd (, 53). These cn be found using smmetr or b noticing tht nother eqution of the circle is r nd then solving the equtions r cos nd r. Arc Length To find the length of polr curve r f, b, we regrd s prmeter nd write the prmetric equtions of the curve s r cos f cos r sin f sin Using the Product Rule nd differentiting with respect to, we obtin d dr d cos r sin d d dr sin r cos d d

34 668 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES so, using cos sin, we hve dr d d d d cos d r dr dr d d dr r sin r dr d sin cos r cos Assuming tht f is continuous, we cn use Theorem..5 to write the rc length s L b d d d cos sin r sin d d d Therefore the length of curve with polr eqution r f, b, is 5 L b r d dr d FIGURE 8 r=+sin v EXAMPLE 4 Find the length of the crdioid r sin. SLUTIN The crdioid is shown in Figure 8. (We sketched it in Emple 7 in Section.3.) Its full length is given b the prmeter intervl, so Formul 5 gives L r d dr s sin d d s sin cos d We could evlute this integrl b multipling nd dividing the integrnd b s sin, or we could use computer lgebr sstem. In n event, we find tht the length of the crdioid is L 8..4 Eercises 4 Find the re of the region tht is bounded b the given curve nd lies in the specified sector.. r e 4, 5 8 Find the re of the shded region r cos, 6 3. r 9 sin, r, 4. r tn, 6 3 r=œ r=+cos ; Grphing clcultor or computer required. Homework Hints vilble t stewrtclculus.com

35 SECTIN.4 AREAS AND LENGTHS IN PLAR CRDINATES Sketch the curve nd find the re tht it encloses. ; 3 6 Grph the curve nd find the re tht it encloses. 7 Find the re of the region enclosed b one loop of the curve.. r sin (inner loop). Find the re enclosed b the loop of the strophoid r cos sec. 3 8 Find the re of the region tht lies inside the first curve nd outside the second curve. 3. r cos, r 4. r sin, 5. r 8 cos, 6. r sin, 7. r 3 cos, 8. r 3sin, 9 34 Find the re of the region tht lies inside both curves. 9. r s3 cos, 3. r cos, 3. r sin, r=4+3 sin 3. r 3 cos, 33. r sin, 34. r sin, r b cos,, b r=sin 9. r sin. r sin. r 3 cos. r 4 3 sin 3. r sin 4 4. r 3 cos 4 5. r s cos 5 6. r 5 sin 6 7. r 4 cos 3 8. r sin 9. r sin 4. r sin 5 r r 3sin r cos r sin r sin r cos r cos r 3 sin r cos r 35. Find the re inside the lrger loop nd outside the smller loop of the limçon r cos. 36. Find the re between lrge loop nd the enclosed smll loop of the curve r cos Find ll points of intersection of the given curves. 37. r sin, 38. r cos, 39. r sin, r 4. r cos 3, 4. r sin, 4. r sin, ; 43. The points of intersection of the crdioid r sin nd the spirl loop r,, cn t be found ectl. Use grphing device to find the pproimte vlues of t which the intersect. Then use these vlues to estimte the re tht lies inside both curves. 44. When recording live performnces, sound engineers often use microphone with crdioid pickup pttern becuse it suppresses noise from the udience. Suppose the microphone is plced 4 m from the front of the stge (s in the figure) nd the boundr of the optiml pickup region is given b the crdioid r 8 8 sin, where r is mesured in meters nd the microphone is t the pole. The musicins wnt to know the re the will hve on stge within the optiml pickup rnge of the microphone. Answer their question Find the ect length of the polr curve. 45. r cos, 46. r 5, 47. r, 48. r cos r 3 sin r sin r sin 3 r sin stge r cos udience m 4 m microphone ; 49 5 Find the ect length of the curve. Use grph to determine the prmeter intervl. 49. r cos r cos

36 67 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES 5 54 Use clcultor to find the length of the curve correct to four deciml plces. If necessr, grph the curve to determine the prmeter intervl. 5. ne loop of the curve r cos 5. r tn, r sin6 sin 54. r sin4 55. () Use Formul..6 to show tht the re of the surfce generted b rotting the polr curve r f b (where f is continuous nd b ) bout the polr is is S b r sin r d dr d (b) Use the formul in prt () to find the surfce re generted b rotting the lemniscte r cos bout the polr is. 56. () Find formul for the re of the surfce generted b rotting the polr curve r f, b (where f is continuous nd b ), bout the line. (b) Find the surfce re generted b rotting the lemniscte r cos bout the line..5 Conic Sections In this section we give geometric definitions of prbols, ellipses, nd hperbols nd derive their stndrd equtions. The re clled conic sections, or conics, becuse the result from intersecting cone with plne s shown in Figure. ellipse prbol hperbol FIGURE Conics Prbols is focus verte FIGURE F prbol directri A prbol is the set of points in plne tht re equidistnt from fied point F (clled the focus) nd fied line (clled the directri). This definition is illustrted b Figure. Notice tht the point hlfw between the focus nd the directri lies on the prbol; it is clled the verte. The line through the focus perpendiculr to the directri is clled the is of the prbol. In the 6th centur Glileo showed tht the pth of projectile tht is shot into the ir t n ngle to the ground is prbol. Since then, prbolic shpes hve been used in designing utomobile hedlights, reflecting telescopes, nd suspension bridges. (See Problem on pge 7 for the reflection propert of prbols tht mkes them so useful.) We obtin prticulrl simple eqution for prbol if we plce its verte t the origin nd its directri prllel to the -is s in Figure 3. If the focus is the point, p, then the directri hs the eqution p. If P, is n point on the prbol,

37 SECTIN.5 CNIC SECTINS 67 FIGURE 3 F(, p) P(, ) =_p p then the distnce from P to the focus is PF s p p nd the distnce from P to the directri is. (Figure 3 illustrtes the cse where p.) The defining propert of prbol is tht these distnces re equl: s p p We get n equivlent eqution b squring nd simplifing: p p p p p p p 4p An eqution of the prbol with focus, p nd directri p is 4p If we write 4p, then the stndrd eqution of prbol becomes. It opens upwrd if p nd downwrd if p [see Figure 4, prts () nd (b)]. The grph is smmetric with respect to the -is becuse is unchnged when is replced b. (, p) =_p (, p) =_p =_p ( p, ) (p, ) =_p () =4p, p> (b) =4p, p< (c) =4p, p> (d) =4p, p< FIGURE 4 If we interchnge nd in, we obtin += _, 5 FIGURE 5 = 5 4p which is n eqution of the prbol with focus p, nd directri p. (Inter chnging nd mounts to reflecting bout the digonl line.) The prbol opens to the right if p nd to the left if p [see Figure 4, prts (c) nd (d)]. In both cses the grph is smmetric with respect to the -is, which is the is of the prbol. EXAMPLE Find the focus nd directri of the prbol nd sketch the grph. SLUTIN If we write the eqution s nd compre it with Eqution, we see tht 4p, so p 5. Thus the focus is p, ( 5 nd the directri is 5, ). The sketch is shown in Figure 5.

38 67 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES Ellipses An ellipse is the set of points in plne the sum of whose distnces from two fied points F nd F is constnt (see Figure 6). These two fied points re clled the foci (plurl of focus). ne of Kepler s lws is tht the orbits of the plnets in the solr sstem re ellipses with the sun t one focus. P P(, ) F F F (_c, ) F (c, ) FIGURE 6 FIGURE 7 In order to obtin the simplest eqution for n ellipse, we plce the foci on the -is t the points c, nd c, s in Figure 7 so tht the origin is hlfw between the foci. Let the sum of the distnces from point on the ellipse to the foci be. Then P, is point on the ellipse when PF PF tht is, s c s c or s c s c Squring both sides, we hve c c 4 4s c c c which simplifies to We squre gin: which becomes s c c c c 4 c c c c From tringle F F P in Figure 7 we see tht c, so c nd therefore c. For convenience, let b c. Then the eqution of the ellipse becomes b b or, if both sides re divided b b, (_, ) (, b) b (, ) (_c, ) c (c, ) (, _b) FIGURE 8 + =, b@ 3 b Since b c, it follows tht b. The -intercepts re found b setting. Then, or, so. The corresponding points, nd, re clled the vertices of the ellipse nd the line segment joining the vertices is clled the mjor is. To find the -intercepts we set nd obtin b, so b. The line segment joining, b nd, b is the minor is. Eqution 3 is unchnged if is replced b or is replced b, so the ellipse is smmetric bout both es. Notice tht if the foci coincide, then c, so b nd the ellipse becomes circle with rdius r b. We summrize this discussion s follows (see lso Figure 8).

39 SECTIN.5 CNIC SECTINS 673 (, ) (, c) 4 The ellipse b b hs foci c,, where c b, nd vertices,. (_b, ) (, _c) (b, ) If the foci of n ellipse re locted on the -is t, c, then we cn find its eqution b interchnging nd in 4. (See Figure 9.) FIGURE 9 + =, b (, _) 5 The ellipse b b hs foci, c, where c b, nd vertices,. (, 3) v EXAMPLE Sketch the grph of nd locte the foci. SLUTIN Divide both sides of the eqution b 44: (_4, ) {_œ 7, } (4, ) {œ 7, } (, _3) 6 9 The eqution is now in the stndrd form for n ellipse, so we hve 6, b 9, 4, nd b 3. The -intercepts re 4 nd the -intercepts re 3. Also, c b 7, so c s7 nd the foci re (s7, ). The grph is sketched in Figure. FIGURE 9 +6 =44 v EXAMPLE 3 Find n eqution of the ellipse with foci, nd vertices, 3. SLUTIN Using the nottion of 5, we hve c nd 3. Then we obtin b c 9 4 5, so n eqution of the ellipse is 5 9 Another w of writing the eqution is FIGURE P is on the hperbol when PF - PF =. P(, ) F (_c, ) F (c, ) Like prbols, ellipses hve n interesting reflection propert tht hs prcticl consequences. If source of light or sound is plced t one focus of surfce with ellipticl cross-sections, then ll the light or sound is reflected off the surfce to the other focus (see Eercise 65). This principle is used in lithotrips, tretment for kidne stones. A reflector with ellipticl cross-section is plced in such w tht the kidne stone is t one focus. High-intensit sound wves generted t the other focus re reflected to the stone nd destro it without dmging surrounding tissue. The ptient is spred the trum of surger nd recovers within few ds. Hperbols A hperbol is the set of ll points in plne the difference of whose distnces from two fied points F nd F (the foci) is constnt. This definition is illustrted in Figure. Hperbols occur frequentl s grphs of equtions in chemistr, phsics, biolog, nd economics (Bole s Lw, hm s Lw, suppl nd demnd curves). A prticulrl signifi-

40 674 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES cnt ppliction of hperbols is found in the nvigtion sstems developed in World Wrs I nd II (see Eercise 5). Notice tht the definition of hperbol is similr to tht of n ellipse; the onl chnge is tht the sum of distnces hs become difference of distnces. In fct, the derivtion of the eqution of hperbol is lso similr to the one given erlier for n ellipse. It is left s Eercise 5 to show tht when the foci re on the -is t c, nd the difference of distnces is, then the eqution of the hperbol is PF PF 6 b (_, ) b =_ (_c, ) FIGURE - b@ b = (, ) (c, ) where c b. Notice tht the -intercepts re gin nd the points, nd, re the vertices of the hperbol. But if we put in Eqution 6 we get b, which is impossible, so there is no -intercept. The hperbol is smmetric with respect to both es. To nlze the hperbol further, we look t Eqution 6 nd obtin This shows tht, so. Therefore we hve or. This mens tht the hperbol consists of two prts, clled its brnches. When we drw hperbol it is useful to first drw its smptotes, which re the dshed lines b nd b shown in Figure. Both brnches of the hperbol pproch the smptotes; tht is, the come rbitrril close to the smptotes. [See Eercise 73 in Section 4.5, where these lines re shown to be slnt smptotes.] 7 The hperbol b s b hs foci c,, where c b, vertices,, nd smptotes b. (, c) If the foci of hperbol re on the -is, then b reversing the roles of nd we obtin the following informtion, which is illustrted in Figure 3. =_ b = b (, ) 8 The hperbol (, _) b FIGURE 3 - b@ (, _c) hs foci, c, where c b, vertices,, nd smptotes b. EXAMPLE 4 its grph. Find the foci nd smptotes of the hperbol nd sketch

41 SECTIN.5 CNIC SECTINS =_ 4 3 = 4 SLUTIN If we divide both sides of the eqution b 44, it becomes (_5, ) (_4, ) (4, ) (5, ) 6 9 which is of the form given in 7 with 4 nd b 3. Since c 6 9 5, the foci re 5,. The smptotes re the lines 3 4 nd 3 4. The grph is shown in Figure 4. FIGURE =44 EXAMPLE 5 Find the foci nd eqution of the hperbol with vertices, nd smptote. SLUTIN From 8 nd the given informtion, we see tht nd b. Thus b nd c b 5 4. The foci re (, s5) nd the eqution of the hperbol is 4 Shifted Conics As discussed in Appendi C, we shift conics b tking the stndrd equtions,, 4, 5, 7, nd 8 nd replcing nd b h nd k. EXAMPLE 6 Find n eqution of the ellipse with foci,, 4, nd vertices,, 5,. SLUTIN The mjor is is the line segment tht joins the vertices,, 5, nd hs length 4, so. The distnce between the foci is, so c. Thus b c 3. Since the center of the ellipse is 3,, we replce nd in 4 b 3 nd to obtin 3 -=_ (-4) s the eqution of the ellipse. v EXAMPLE 7 3 Sketch the conic nd find its foci. SLUTIN We complete the squres s follows: 4 3 (4, 4) (4, ) (4, _) 3 -= (-4) FIGURE = This is in the form 8 ecept tht nd re replced b 4 nd. Thus 9, b 4, nd c 3. The hperbol is shifted four units to the right nd one unit upwrd. The foci re (4, s3 ) nd (4, s3 ) nd the vertices re 4, 4 nd 4,. The smptotes re 3 4. The hperbol is sketched in Figure 5.

42 676 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.5 Eercises 8 Find the verte, focus, nd directri of the prbol nd sketch its grph Find n eqution of the prbol. Then find the focus nd directri Find the vertices nd foci of the ellipse nd sketch its grph _ Find n eqution of the ellipse. Then find its foci Identif the tpe of conic section whose eqution is given nd find the vertices nd foci Find n eqution for the conic tht stisfies the given conditions. 3. Prbol, verte,, focus, 3. Prbol, focus,, directri Prbol, focus 4,, directri 34. Prbol, focus 3, 6, verte 3, 35. Prbol, verte, 3, verticl is, pssing through, Prbol, horizontl is, pssing through,,,, nd 3, 37. Ellipse, foci,, vertices 5, 38. Ellipse, foci, 5, vertices, Ellipse, foci,,, 6, vertices,,, 8 4. Ellipse, foci,, 8,, verte 9, 4. Ellipse, center, 4, verte,, focus, 6 4. Ellipse, foci 4,, pssing through 4, Hperbol, vertices 3,, foci 5, 44. Hperbol, vertices,, foci, Hperbol, vertices 3, 4, 3, 6, foci 3, 7, 3, Find the vertices, foci, nd smptotes of the hperbol nd sketch its grph Hperbol, vertices,, 7,, foci,, 8, 47. Hperbol, vertices 3,, smptotes 48. Hperbol, foci,,, 8, smptotes 3 nd 5. Homework Hints vilble t stewrtclculus.com

43 SECTIN.5 CNIC SECTINS The point in lunr orbit nerest the surfce of the moon is clled perilune nd the point frthest from the surfce is clled polune. The Apollo spcecrft ws plced in n ellipticl lunr orbit with perilune ltitude km nd polune ltitude 34 km (bove the moon). Find n eqution of this ellipse if the rdius of the moon is 78 km nd the center of the moon is t one focus. 5. A cross-section of prbolic reflector is shown in the figure. The bulb is locted t the focus nd the opening t the focus is cm. () Find n eqution of the prbol. (b) Find the dimeter of the opening CD, cm from the verte. V 5 cm cm F 5 cm 5. In the LRAN (Lng RAnge Nvigtion) rdio nvigtion sstem, two rdio sttions locted t A nd B trnsmit simul tneous signls to ship or n ircrft locted t P. The onbord computer converts the time difference in receiving these signls into distnce difference PA PB, nd this, ccording to the definition of hperbol, loctes the ship or ircrft on one brnch of hperbol (see the figure). Suppose tht sttion B is locted 4 mi due est of sttion A on costline. A ship received the signl from B micro seconds (s) before it received the signl from A. () Assuming tht rdio signls trvel t speed of 98 fts, find n eqution of the hperbol on which the ship lies. (b) If the ship is due north of B, how fr off the costline is the ship? A 5. Use the definition of hperbol to derive Eqution 6 for hperbol with foci c, nd vertices,. 53. Show tht the function defined b the upper brnch of the hperbol b is concve upwrd. A B costline 4 mi trnsmitting sttions C D P B 54. Find n eqution for the ellipse with foci, nd, nd mjor is of length Determine the tpe of curve represented b the eqution k k 6 in ech of the following cses: () k 6, (b) k 6, nd (c) k. (d) Show tht ll the curves in prts () nd (b) hve the sme foci, no mtter wht the vlue of k is. 56. () Show tht the eqution of the tngent line to the prbol 4p t the point, cn be written s p (b) Wht is the -intercept of this tngent line? Use this fct to drw the tngent line. 57. Show tht the tngent lines to the prbol 4p drwn from n point on the directri re perpendiculr. 58. Show tht if n ellipse nd hperbol hve the sme foci, then their tngent lines t ech point of intersection re perpendiculr. 59. Use prmetric equtions nd Simpson s Rule with n 8 to estimte the circumference of the ellipse The plnet Pluto trvels in n ellipticl orbit round the sun (t one focus). The length of the mjor is is.8 km nd the length of the minor is is.4 km. Use Simpson s Rule with n to estimte the distnce trveled b the plnet during one complete orbit round the sun. 6. Find the re of the region enclosed b the hperbol b nd the verticl line through focus. 6. () If n ellipse is rotted bout its mjor is, find the volume of the resulting solid. (b) If it is rotted bout its minor is, find the resulting volume. 63. Find the centroid of the region enclosed b the -is nd the top hlf of the ellipse () Clculte the surfce re of the ellipsoid tht is generted b rotting n ellipse bout its mjor is. (b) Wht is the surfce re if the ellipse is rotted bout its minor is? 65. Let P be point on the ellipse b, with foci nd nd let nd be the ngles between the lines F F

44 678 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES PF, PF nd the ellipse s shown in the figure. Prove tht. This eplins how whispering glleries nd litho trips work. Sound coming from one focus is reflected nd psses through the other focus. [Hint: Use the formul in Problem 9 on pge 7 to show tht tn tn.] P(, ) å hperbol. It shows tht light imed t focus F of hperbolic mirror is reflected towrd the other focus.) F P å F F F F + b@ P 66. Let P be point on the hperbol b, with foci F nd F nd let nd be the ngles between the lines PF, PF nd the hperbol s shown in the figure. Prove tht. (This is the reflection propert of the F F.6 Conic Sections in Polr Coordintes In the preceding section we defined the prbol in terms of focus nd directri, but we defined the ellipse nd hperbol in terms of two foci. In this section we give more unified tretment of ll three tpes of conic sections in terms of focus nd directri. Furthermore, if we plce the focus t the origin, then conic section hs simple polr eqution, which provides convenient description of the motion of plnets, stellites, nd comets. Theorem Let F be fied point (clled the focus) nd l be fied line (clled the directri) in plne. Let e be fied positive number (clled the eccentricit). The set of ll points P in the plne such tht PF Pl (tht is, the rtio of the distnce from F to the distnce from l is the constnt e) is conic section. The conic is e () (b) (c) n ellipse if e prbol if e hperbol if e PF Pl PRF Notice tht if the eccentricit is e, then nd so the given condition simpl becomes the definition of prbol s given in Section.5.

45 SECTIN.6 CNIC SECTINS IN PLAR CRDINATES 679 F P r r cos C d l (directri) =d Let us plce the focus F t the origin nd the directri prllel to the -is nd d units to the right. Thus the directri hs eqution d nd is perpendiculr to the polr is. If the point P hs polr coordintes r,, we see from Figure tht PF r Pl PF Pl e PF e Pl d r cos Thus the condition, or, becomes r ed r cos If we squre both sides of this polr eqution nd convert to rectngulr coordintes, we get e d e d d FIGURE or e de e d After completing the squre, we hve e d e e e d 3 e If e, we recognize Eqution 3 s the eqution of n ellipse. In fct, it is of the form h b where 4 h e d e e d e b e d e In Section.5 we found tht the foci of n ellipse re t distnce c from the center, where c b e 4 d 5 e This shows tht c e d h e nd confirms tht the focus s defined in Theorem mens the sme s the focus defined in Section.5. It lso follows from Equtions 4 nd 5 tht the eccentricit is given b e c If e, then e nd we see tht Eqution 3 represents hperbol. Just s we did before, we could rewrite Eqution 3 in the form nd see tht h b e c where c b

46 68 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES B solving Eqution for r, we see tht the polr eqution of the conic shown in Figure cn be written s ed r e cos If the directri is chosen to be to the left of the focus s d, or if the directri is chosen to be prllel to the polr is s d, then the polr eqution of the conic is given b the following theorem, which is illustrted b Figure. (See Eercises 3.) =d directri =_d directri =d directri F F F F =_ d directri () r= ed +e cos (b) r= ed -e cos (c) r= ed +e sin (d) r= ed -e sin FIGURE Polr equtions of conics 6 Theorem A polr eqution of the form r ed e cos or r ed e sin represents conic section with eccentricit e. The conic is n ellipse if e, prbol if e, or hperbol if e. v EXAMPLE Find polr eqution for prbol tht hs its focus t the origin nd whose directri is the line 6. SLUTIN Using Theorem 6 with e nd d 6, nd using prt (d) of Figure, we see tht the eqution of the prbol is r 6 sin v EXAMPLE A conic is given b the polr eqution r 3 cos Find the eccentricit, identif the conic, locte the directri, nd sketch the conic. SLUTIN Dividing numertor nd denomintor b 3, we write the eqution s r 3 3 cos

47 =_5 (directri) FIGURE 3 (, π) r= 3- cos focus (, ) SECTIN.6 CNIC SECTINS IN PLAR CRDINATES 68 From Theorem 6 we see tht this represents n ellipse with e. Since ed 3 3, we hve so the directri hs Crtesin eqution 5. When, r ; when, r. So the vertices hve polr coordintes, nd,. The ellipse is sketched in Figure 3. EXAMPLE 3 Sketch the conic r. 4sin SLUTIN Writing the eqution in the form 3 d e 3 5 r we see tht the eccentricit is e nd the eqution therefore represents hperbol. Since ed 6, d 3 nd the directri hs eqution 3. The vertices occur when nd 3, so the re, nd 6, 3 6,. It is lso useful to plot the -intercepts. These occur when, ; in both cses r 6. For dditionl ccurc we could drw the smptotes. Note tht r l when sin l or nd sin when sin. Thus the smptotes re prllel to the rs 76 nd 6. The hperbol is sketched in Figure sin π 6, π, =3 (directri) FIGURE 4 r= +4 sin (6, π) (6, ) focus r= 3- cos( -π/4) When rotting conic sections, we find it much more convenient to use polr equtions thn Crtesin equtions. We just use the fct (see Eercise 73 in Section.3) tht the grph of r f is the grph of r f rotted counterclockwise bout the origin through n ngle. v EXAMPLE 4 If the ellipse of Emple is rotted through n ngle 4 bout the origin, find polr eqution nd grph the resulting ellipse. SLUTIN We get the eqution of the rotted ellipse b replcing with 4 in the eqution given in Emple. So the new eqution is _5 5 FIGURE 5 _6 r= 3- cos r 3 cos 4 We use this eqution to grph the rotted ellipse in Figure 5. Notice tht the ellipse hs been rotted bout its left focus.

48 68 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES In Figure 6 we use computer to sketch number of conics to demonstrte the effect of vring the eccentricit e. Notice tht when e is close to the ellipse is nerl circulr, wheres it becomes more elongted s e l. When e, of course, the conic is prbol. e=. e=.5 e=.68 e=.86 e=.96 FIGURE 6 e= e=. e=.4 e=4 Kepler s Lws In 69 the Germn mthemticin nd stronomer Johnnes Kepler, on the bsis of huge mounts of stronomicl dt, published the following three lws of plnetr motion. Kepler s Lws. A plnet revolves round the sun in n ellipticl orbit with the sun t one focus.. The line joining the sun to plnet sweeps out equl res in equl times. 3. The squre of the period of revolution of plnet is proportionl to the cube of the length of the mjor is of its orbit. Although Kepler formulted his lws in terms of the motion of plnets round the sun, the ppl equll well to the motion of moons, comets, stellites, nd other bodies tht orbit subject to single grvittionl force. In Section 3.4 we will show how to deduce Kepler s Lws from Newton s Lws. Here we use Kepler s First Lw, together with the polr eqution of n ellipse, to clculte quntities of interest in stronom. For purposes of stronomicl clcultions, it s useful to epress the eqution of n ellipse in terms of its eccentricit e nd its semimjor is. We cn write the distnce d from the focus to the directri in terms of if we use 4 : e d e? d e? d e e e So ed e. If the directri is d, then the polr eqution is r ed e cos e e cos

49 SECTIN.6 CNIC SECTINS IN PLAR CRDINATES The polr eqution of n ellipse with focus t the origin, semimjor is, eccentricit e, nd directri d cn be written in the form r e e cos plnet r sun phelion perihelion The positions of plnet tht re closest to nd frthest from the sun re clled its perihelion nd phelion, respectivel, nd correspond to the vertices of the ellipse. (See Figure 7.) The distnces from the sun to the perihelion nd phelion re clled the perihelion distnce nd phelion distnce, respectivel. In Figure the sun is t the focus F, so t perihelion we hve nd, from Eqution 7, FIGURE 7 r e e cos e e e e Similrl, t phelion nd r e. 8 The perihelion distnce from plnet to the sun is e nd the phelion distnce is e. EXAMPLE 5 () Find n pproimte polr eqution for the ellipticl orbit of the erth round the sun (t one focus) given tht the eccentricit is bout.7 nd the length of the mjor is is bout.99 8 km. (b) Find the distnce from the erth to the sun t perihelion nd t phelion. SLUTIN () The length of the mjor is is.99 8, so We re given tht e.7 nd so, from Eqution 7, n eqution of the erth s orbit round the sun is or, pproimtel, r e e cos cos r cos (b) From 8, the perihelion distnce from the erth to the sun is nd the phelion distnce is e km e km

50 684 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.6 Eercises 8 Write polr eqution of conic with the focus t the origin nd the given dt.. Ellipse, eccentricit, directri 4. Prbol, directri 3 3. Hperbol, eccentricit.5, directri 4. Hperbol, eccentricit 3, directri 3 5. Prbol, verte 4, 3 6. Ellipse, eccentricit.8, verte, 7. Ellipse, eccentricit, directri r 4 sec 8. Hperbol, eccentricit 3, directri r 6 csc 9 6 () Find the eccentricit, (b) identif the conic, (c) give n eqution of the directri, nd (d) sketch the conic r. r 5 4 sin. r. r 3 3 sin 9 3. r 4. r 6 cos 3 5. r 6. r 4 8 cos 3 cos 3 cos sin 5 6 sin ; 7. () Find the eccentricit nd directri of the conic r sin nd grph the conic nd its directri. (b) If this conic is rotted counterclockwise bout the origin through n ngle 34, write the resulting eqution nd grph its curve. ; 8. Grph the conic r 45 6 cos nd its directri. Also grph the conic obtined b rotting this curve bout the origin through n ngle 3. ; 9. Grph the conics r e e cos with e.4,.6,.8, nd. on common screen. How does the vlue of e ffect the shpe of the curve? ;. () Grph the conics r ed e sin for e nd vrious vlues of d. How does the vlue of d ffect the shpe of the conic? (b) Grph these conics for d nd vrious vlues of e. How does the vlue of e ffect the shpe of the conic?. Show tht conic with focus t the origin, eccentricit e, nd directri d hs polr eqution r ed e cos. Show tht conic with focus t the origin, eccentricit e, nd directri d hs polr eqution r ed e sin 3. Show tht conic with focus t the origin, eccentricit e, nd directri d hs polr eqution r ed e sin 4. Show tht the prbols r c cos nd r d cos intersect t right ngles. 5. The orbit of Mrs round the sun is n ellipse with eccentricit.93 nd semimjor is.8 8 km. Find polr eqution for the orbit. 6. Jupiter s orbit hs eccentricit.48 nd the length of the mjor is is.56 9 km. Find polr eqution for the orbit. 7. The orbit of Hlle s comet, lst seen in 986 nd due to return in 6, is n ellipse with eccentricit.97 nd one focus t the sun. The length of its mjor is is 36.8 AU. [An stronomicl unit (AU) is the men distnce between the erth nd the sun, bout 93 million miles.] Find polr eqution for the orbit of Hlle s comet. Wht is the mimum distnce from the comet to the sun? 8. The Hle-Bopp comet, discovered in 995, hs n ellipticl orbit with eccentricit.995 nd the length of the mjor is is AU. Find polr eqution for the orbit of this comet. How close to the sun does it come? Den Ketelsen 9. The plnet Mercur trvels in n ellipticl orbit with eccentricit.6. Its minimum distnce from the sun is km. Find its mimum distnce from the sun. 3. The distnce from the plnet Pluto to the sun is km t perihelion nd km t phelion. Find the eccentricit of Pluto s orbit. 3. Using the dt from Eercise 9, find the distnce trveled b the plnet Mercur during one complete orbit round the sun. (If our clcultor or computer lgebr sstem evlutes definite integrls, use it. therwise, use Simpson s Rule.) ; Grphing clcultor or computer required. Homework Hints vilble t stewrtclculus.com