PARAMETRIC EQUATIONS AND POLAR COORDINATES

Transcription

1 PARAMETRIC EQUATINS AND PLAR CRDINATES Parametric equations and polar coordinates enable us to describe a great variet of new curves some practical, some beautiful, some fanciful, some strange. So far we have described plane curves b giving as a function of f or as a function of t or b giving a relation between and that defines implicitl as a function of f,. In this chapter we discuss two new methods for describing curves. Some curves, such as the ccloid, are best handled when both and are given in terms of a third variable t called a parameter f t, tt. ther curves, such as the cardioid, have their most convenient description when we use a new coordinate sstem, called the polar coordinate sstem. 6

2 . CURVES DEFINED BY PARAMETRIC EQUATINS FIGURE C (, )={ f(t), g(t)} Imagine that a particle moves along the curve C shown in Figure. It is impossible to describe C b an equation of the form f because C fails the Vertical Line Test. But the - and -coordinates of the particle are functions of time and so we can write f t and tt. Such a pair of equations is often a convenient wa of describing a curve and gives rise to the following definition. Suppose that and are both given as functions of a third variable t (called a parameter) b the equations f t tt (called parametric equations). Each value of t determines a point,, which we can plot in a coordinate plane. As t varies, the point, f t, tt varies and traces out a curve C, which we call a parametric curve. The parameter t does not necessaril represent time and, in fact, we could use a letter other than t for the parameter. But in man applications of parametric curves, t does denote time and therefore we can interpret, f t, tt as the position of a particle at time t. EXAMPLE Sketch and identif the curve defined b the parametric equations t t t SLUTIN Each value of t gives a point on the curve, as shown in the table. For instance, if t, then, and so the corresponding point is,. In Figure we plot the points, determined b several values of the parameter and we join them to produce a curve. t t= t= t= (, ) t= t=_ t= 8 t=_ FIGURE N This equation in and describes where the particle has been, but it doesn t tell us when the particle was at a particular point. The parametric equations have an advantage the tell us when the particle was at a point. The also indicate the direction of the motion. A particle whose position is given b the parametric equations moves along the curve in the direction of the arrows as t increases. Notice that the consecutive points marked on the curve appear at equal time intervals but not at equal distances. That is because the particle slows down and then speeds up as t increases. It appears from Figure that the curve traced out b the particle ma be a parabola. This can be confirmed b eliminating the parameter t as follows. We obtain t from the second equation and substitute into the first equation. This gives t t and so the curve represented b the given parametric equations is the parabola. M 6

3 6 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES (8, 5) No restriction was placed on the parameter t in Eample, so we assumed that t could be an real number. But sometimes we restrict t to lie in a finite interval. For instance, the parametric curve t t t t (, ) FIGURE shown in Figure is the part of the parabola in Eample that starts at the point, and ends at the point 8, 5. The arrowhead indicates the direction in which the curve is traced as t increases from to. In general, the curve with parametric equations f t tt a t b has initial point f a, ta and terminal point f b, tb. V EXAMPLE What curve is represented b the following parametric equations? cos t sin t t SLUTIN If we plot points, it appears that the curve is a circle. We can confirm this impression b eliminating t. bserve that cos t sin t Thus the point, moves on the unit circle. Notice that in this eample the parameter t can be interpreted as the angle (in radians) shown in Figure. As t increases from to, the point, cos t, sin t moves once around the circle in the counterclockwise direction starting from the point,. π t= (cos t, sin t) t=π t t= (, ) t=π FIGURE π t= M t=, π, π EXAMPLE What curve is represented b the given parametric equations? sin t cos t t (, ) SLUTIN Again we have sin t cos t so the parametric equations again represent the unit circle. But as t increases from to, the point, sin t, cos t starts at, and moves twice around the circle in the clockwise direction as indicated in Figure 5. M FIGURE 5 Eamples and show that different sets of parametric equations can represent the same curve. Thus we distinguish between a curve, which is a set of points, and a parametric curve, in which the points are traced in a particular wa.

4 SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS 6 EXAMPLE Find parametric equations for the circle with center h, k and radius r. SLUTIN If we take the equations of the unit circle in Eample and multipl the epressions for and b r, we get r cos t, r sin t. You can verif that these equations represent a circle with radius r and center the origin traced counterclockwise. We now shift h units in the -direction and k units in the -direction and obtain parametric equations of the circle (Figure 6) with center h, k and radius r: h r cos t k r sin t t r (h, k) FIGURE 6 =h+r cos t, =k+r sin t M (_, ) (, ) V EXAMPLE 5 Sketch the curve with parametric equations sin t, sin t. FIGURE 7 SLUTIN bserve that sin t and so the point, moves on the parabola. But note also that, since sin t, we have, so the parametric equations represent onl the part of the parabola for which. Since sin t is periodic, the point, sin t, sin t moves back and forth infinitel often along the parabola from, to,. (See Figure 7.) M TEC Module.A gives an animation of the relationship between motion along a parametric curve f t, tt and motion along the graphs of f and t as functions of t. Clicking on TRIG gives ou the famil of parametric curves =cos t a cos bt c sin dt If ou choose a b c d and click on animate, ou will see how the graphs of cos t and sin t relate to the circle in Eample. If ou choose a b c, d, ou will see graphs as in Figure 8. B clicking on animate or moving the t-slider to the right, ou can see from the color coding how motion along the graphs of cos t and sin t corresponds to motion along the parametric curve, which is called a Lissajous figure. t t FIGURE 8 =cos t =sin t =sin t

5 6 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES GRAPHING DEVICES Most graphing calculators and computer graphing programs can be used to graph curves defined b parametric equations. In fact, it s instructive to watch a parametric curve being drawn b a graphing calculator because the points are plotted in order as the corresponding parameter values increase. EXAMPLE 6 Use a graphing device to graph the curve. SLUTIN If we let the parameter be t, then we have the equations t t t Using these parametric equations to graph the curve, we obtain Figure 9. It would be possible to solve the given equation for as four functions of and graph them individuall, but the parametric equations provide a much easier method. M FIGURE 9 In general, if we need to graph an equation of the form t, we can use the parametric equations tt t 8 Notice also that curves with equations f (the ones we are most familiar with graphs of functions) can also be regarded as curves with parametric equations Graphing devices are particularl useful when sketching complicated curves. For instance, the curves shown in Figures,, and would be virtuall impossible to produce b hand..5 t f t _ _ _.5 _ FIGURE =t+ sin t =t+ cos 5t FIGURE =.5 cos t-cos t =.5 sin t-sin t FIGURE =sin(t+cos t) =cos(t+sin t) ne of the most important uses of parametric curves is in computer-aided design (CAD). In the Laborator Project after Section. we will investigate special parametric curves, called Bézier curves, that are used etensivel in manufacturing, especiall in the automotive industr. These curves are also emploed in specifing the shapes of letters and other smbols in laser printers. THE CYCLID TEC An animation in Module.B shows how the ccloid is formed as the circle moves. EXAMPLE 7 The curve traced out b a point P on the circumference of a circle as the circle rolls along a straight line is called a ccloid (see Figure ). If the circle has radius r and rolls along the -ais and if one position of P is the origin, find parametric equations for the ccloid.

6 SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS 65 P P FIGURE P r P r FIGURE C(r, r) Q T SLUTIN We choose as parameter the angle of rotation of the circle when P is at the origin). Suppose the circle has rotated through radians. Because the circle has been in contact with the line, we see from Figure that the distance it has rolled from the origin is arc PT r T Therefore the center of the circle is Cr, r. Let the coordinates of P be,. Then from Figure we see that T PQ r r sin r sin TC QC r r cos r cos Therefore parametric equations of the ccloid are r sin r cos A ne arch of the ccloid comes from one rotation of the circle and so is described b. Although Equations were derived from Figure, which illustrates the case where, it can be seen that these equations are still valid for other values of (see Eercise 9). Although it is possible to eliminate the parameter from Equations, the resulting Cartesian equation in and is ver complicated and not as convenient to work with as the parametric equations. M ccloid FIGURE 5 P P FIGURE 6 P P B P ne of the first people to stud the ccloid was Galileo, who proposed that bridges be built in the shape of ccloids and who tried to find the area under one arch of a ccloid. Later this curve arose in connection with the brachistochrone problem: Find the curve along which a particle will slide in the shortest time (under the influence of gravit) from a point A to a lower point B not directl beneath A. The Swiss mathematician John Bernoulli, who posed this problem in 696, showed that among all possible curves that join A to B, as in Figure 5, the particle will take the least time sliding from A to B if the curve is part of an inverted arch of a ccloid. The Dutch phsicist Hugens had alread shown that the ccloid is also the solution to the tautochrone problem; that is, no matter where a particle P is placed on an inverted ccloid, it takes the same time to slide to the bottom (see Figure 6). Hugens proposed that pendulum clocks (which he invented) swing in ccloidal arcs because then the pendulum takes the same time to make a complete oscillation whether it swings through a wide or a small arc. FAMILIES F PARAMETRIC CURVES V EXAMPLE 8 Investigate the famil of curves with parametric equations a cos t a tan t sin t What do these curves have in common? How does the shape change as a increases?

7 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device to produce the graphs for the cases a,,.5,.,,.5,, and shown in Figure 7. Notice that all of these curves (ecept the case a ) have two branches, and both branches approach the vertical asmptote a as approaches a from the left or right. a=_ a=_ a=_.5 a=_. a= a=.5 a= a= FIGURE 7 Members of the famil =a+cos t, =a tan t+sin t, all graphed in the viewing rectangle _, b _, When a, both branches are smooth; but when a reaches, the right branch acquires a sharp point, called a cusp. For a between and the cusp turns into a loop, which becomes larger as a approaches. When a, both branches come together and form a circle (see Eample ). For a between and, the left branch has a loop, which shrinks to become a cusp when a. For a, the branches become smooth again, and as a increases further, the become less curved. Notice that the curves with a positive are reflections about the -ais of the corresponding curves with a negative. These curves are called conchoids of Nicomedes after the ancient Greek scholar Nicomedes. He called them conchoids because the shape of their outer branches resembles that of a conch shell or mussel shell. M. EXERCISES Sketch the curve b using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.. st, t t,. cos t, t cos t,. 5 sin t, t,. e t t, e t t, 5 (a) Sketch the curve b using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. 5. t 5, t 6. t, 5 t, t 5 t t t t 7. t, 5 t, t 8. t, 9. st,. t, 8 (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.. sin, cos,. cos, 5 sin,. sin t, csc t,. e t, 5. e t, t t t e t 6. ln t, st, t 7. sinh t, cosh t t t

8 SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS cosh t, 5 sinh t 9 Describe the motion of a particle with position, as t varies in the given interval. 9. cos t, sin t, t. sin t, cos t, t. 5 sin t, cos t,. sin t, cos t, t 5 t 5 7 Use the graphs of f t and tt to sketch the parametric curve f t, tt. Indicate with arrows the direction in which the curve is traced as t increases. 5. t t _ 6.. Suppose a curve is given b the parametric equations f t, tt, where the range of f is, and the range of t is,. What can ou sa about the curve?. Match the graphs of the parametric equations f t and tt in (a) (d) with the parametric curves labeled I IV. Give reasons for our choices. (a) I 7. t t t t (b) t t t t II 8. Match the parametric equations with the graphs labeled I-VI. Give reasons for our choices. (Do not use a graphing device.) (a) t t, t (b) t t, st (c) sin t, sint sin t (d) cos 5t, sin t (e) t sin t, t cos t sin t cos t (f), t t I II III (c) III t t IV V VI (d) IV t t ; 9. Graph the curve 5. ;. Graph the curves 5 and and find their points of intersection correct to one decimal place.

9 68 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES. (a) Show that the parametric equations where t, describe the line segment that joins the points P, and P,. (b) Find parametric equations to represent the line segment from, 7 to,. ;. Use a graphing device and the result of Eercise (a) to draw the triangle with vertices A,, B,, and C, 5. ;. (a) Find parametric equations for the ellipse a b. [Hint: Modif the equations of the circle in Eample.] (b) Use these parametric equations to graph the ellipse when a and b,,, and 8. (c) How does the shape of the ellipse change as b varies? ; 5 6 Use a graphing calculator or computer to reproduce the picture t t. Find parametric equations for the path of a particle that moves along the circle in the manner described. (a) nce around clockwise, starting at, (b) Three times around counterclockwise, starting at, (c) Halfwa around counterclockwise, starting at,. If a and b are fied numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle as the parameter. Then eliminate the parameter and identif the curve. a b P. If a and b are fied numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle as the parameter. The line segment AB is tangent to the larger circle. A a b P B Compare the curves represented b the parametric equations. How do the differ? 7. (a) t, t (b) t 6, t (c) e t, e t 8. (a) t, t (b) cos t, (c) e t, e t 9. Derive Equations for the case.. Let P be a point at a distance d from the center of a circle of radius r. The curve traced out b P as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a biccle wheel.) The ccloid is the special case of a trochoid with d r. Using the same parameter as for the ccloid and, assuming the line is the -ais and when P is at one of its lowest points, show that parametric equations of the trochoid are r d sin r d cos Sketch the trochoid for the cases d r and d r. sec t. A curve, called a witch of Maria Agnesi, consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written as a cot Sketch the curve. =a a a sin. (a) Find parametric equations for the set of all points P as shown in the figure such that P AB. (This curve is called the cissoid of Diocles after the Greek scholar Diocles, who introduced the cissoid as a graphical method for constructing the edge of a cube whose volume is twice that of a given cube.) A C P

10 LABRATRY PRJECT RUNNING CIRCLES ARUND CIRCLES 69 (b) Use the geometric description of the curve to draw a rough sketch of the curve b hand. Check our work b using the parametric equations to graph the curve. ; 5. Suppose that the position of one particle at time t is given b sin t and the position of a second particle is given b cos t (a) Graph the paths of both particles. How man points of intersection are there? (b) Are an of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points. (c) Describe what happens if the path of the second particle is given b cos t sin t t 6. If a projectile is fired with an initial velocit of v meters per second at an angle above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is P a A cos t B sin t =a t t given b the parametric equations v cos t where t is the acceleration due to gravit ( 9.8 ms ). (a) If a gun is fired with and v 5 ms, when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maimum height reached b the bullet? ; (b) Use a graphing device to check our answers to part (a). Then graph the path of the projectile for several other values of the angle to see where it hits the ground. Summarize our findings. (c) Show that the path is parabolic b eliminating the parameter. v sin t tt ; 7. Investigate the famil of curves defined b the parametric equations t, t ct. How does the shape change as c increases? Illustrate b graphing several members of the famil. ; 8. The swallowtail catastrophe curves are defined b the parametric equations ct t, ct t. Graph several of these curves. What features do the curves have in common? How do the change when c increases? ; 9. The curves with equations a sin nt, b cos t are called Lissajous figures. Investigate how these curves var when a, b, and n var. (Take n to be a positive integer.) ; 5. Investigate the famil of curves defined b the parametric equations cos t, sin t sin ct, where c. Start b letting c be a positive integer and see what happens to the shape as c increases. Then eplore some of the possibilities that occur when c is a fraction. L A B R AT R Y P R J E C T ; RUNNING CIRCLES ARUND CIRCLES In this project we investigate families of curves, called hpoccloids and epiccloids, that are generated b the motion of a point on a circle that rolls inside or outside another circle. a C b P (a, ) A. A hpoccloid is a curve traced out b a fied point P on a circle C of radius b as C rolls on the inside of a circle with center and radius a. Show that if the initial position of P is a, and the parameter is chosen as in the figure, then parametric equations of the hpoccloid are a b cos b cos a b b a b sin b sin a b b. Use a graphing device (or the interactive graphic in TEC Module.B) to draw the graphs of hpoccloids with a a positive integer and b. How does the value of a affect the graph? Show that if we take a, then the parametric equations of the hpoccloid reduce to TEC Look at Module.B to see how hpoccloids and epiccloids are formed b the motion of rolling circles. cos sin This curve is called a hpoccloid of four cusps, or an astroid.

11 6 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES. Now tr b and a nd, a fraction where n and d have no common factor. First let n and tr to determine graphicall the effect of the denominator d on the shape of the graph. Then let n var while keeping d constant. What happens when n d?. What happens if b and a is irrational? Eperiment with an irrational number like s or e. Take larger and larger values for and speculate on what would happen if we were to graph the hpoccloid for all real values of. 5. If the circle C rolls on the outside of the fied circle, the curve traced out b P is called an epiccloid. Find parametric equations for the epiccloid. 6. Investigate the possible shapes for epiccloids. Use methods similar to Problems.. CALCULUS WITH PARAMETRIC CURVES Having seen how to represent curves b parametric equations, we now appl the methods of calculus to these parametric curves. In particular, we solve problems involving tangents, area, arc length, and surface area. TANGENTS In the preceding section we saw that some curves defined b parametric equations f t and tt can also be epressed, b eliminating the parameter, in the form F. (See Eercise 67 for general conditions under which this is possible.) If we substitute f t and tt in the equation F, we get tt F f t and so, if t, F, and f are differentiable, the Chain Rule gives tt F f tf t F f t If f t, we can solve for F: F tt f t Since the slope of the tangent to the curve F at, F is F, Equation enables us to find tangents to parametric curves without having to eliminate the parameter. Using Leibniz notation, we can rewrite Equation in an easil remembered form: N If we think of a parametric curve as being traced out b a moving particle, then ddt and ddt are the vertical and horizontal velocities of the particle and Formula sas that the slope of the tangent is the ratio of these velocities. d d d dt d dt if d dt It can be seen from Equation that the curve has a horizontal tangent when ddt (provided that ddt ) and it has a vertical tangent when ddt (provided that ddt ). This information is useful for sketching parametric curves.

12 SECTIN. CALCULUS WITH PARAMETRIC CURVES 6 Note that d d d dt d dt As we know from Chapter, it is also useful to consider d d. This can be found b replacing b dd in Equation : d d dt d d d d d d d d dt EXAMPLE A curve C is defined b the parametric equations t, t t. (a) Show that C has two tangents at the point (, ) and find their equations. (b) Find the points on C where the tangent is horizontal or vertical. (c) Determine where the curve is concave upward or downward. (d) Sketch the curve. SLUTIN (a) Notice that t t tt when t or t s. Therefore the point, on C arises from two values of the parameter, t s and t s. This indicates that C crosses itself at,. Since d ddt d ddt t t t t the slope of the tangent when t s is dd 6(s ) s, so the equations of the tangents at, are s and s t=_ (, ) t= (, _) FIGURE =œ (-) (, ) =_ œ (-) (b) C has a horizontal tangent when dd, that is, when ddt and ddt. Since ddt t, this happens when t, that is, t. The corresponding points on C are, and (, ). C has a vertical tangent when ddt t, that is, t. (Note that ddt there.) The corresponding point on C is (, ). (c) To determine concavit we calculate the second derivative: d d d dt d d t t d t t dt Thus the curve is concave upward when t and concave downward when t. (d) Using the information from parts (b) and (c), we sketch C in Figure. M V EXAMPLE (a) Find the tangent to the ccloid r sin, r cos at the point where. (See Eample 7 in Section..) (b) At what points is the tangent horizontal? When is it vertical? SLUTIN (a) The slope of the tangent line is d d dd dd r sin r cos sin cos

13 6 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES When, we have r sin r s r cos r and d d sin cos s s Therefore the slope of the tangent is s and its equation is r s r The tangent is sketched in Figure. rs or s r s (_πr, r) (πr, r) (πr, r) (5πr, r) π = FIGURE πr πr (b) The tangent is horizontal when dd, which occurs when sin and cos, that is,, n an integer. The corresponding point on the ccloid is n r, r. When, both and are. It appears from the graph that there are vertical tangents at those points. We can verif this b using l Hospital s Rule as follows: n d lim ln d A similar computation shows that ddl as l n, so indeed there are vertical tangents when, that is, when nr. M n n dd lim ln dd sin cos lim cos ln sin AREAS N The limits of integration for t are found as usual with the Substitution Rule. When a, t is either. When b, t is the remaining value. or We know that the area under a curve F from a to b is A b F d, where a F. If the curve is traced out once b the parametric equations f t and tt,, then we can calculate an area formula b using the Substitution Rule for Definite Integrals as follows: A or tt f t dt b d tt f t dt t a V EXAMPLE Find the area under one arch of the ccloid (See Figure.) r sin r cos

14 SECTIN. CALCULUS WITH PARAMETRIC CURVES 6 FIGURE πr N The result of Eample sas that the area under one arch of the ccloid is three times the area of the rolling circle that generates the ccloid (see Eample 7 in Section.). Galileo guessed this result but it was first proved b the French mathematician Roberval and the Italian mathematician Torricelli. SLUTIN ne arch of the ccloid is given b. Using the Substitution Rule with r cos and d r cos d, we have ARC LENGTH We alread know how to find the length L of a curve C given in the form F, a b. Formula 8.. sas that if F is continuous, then A r r d r cos r cos d r cos d r cos cos d [ cos cos ] d r [ r ( sin sin ] ) r b L d d a d M Suppose that C can also be described b the parametric equations f t and tt,, where ddt f t. This means that C is traversed once, from left to right, as t increases from to and f a, f b. Putting Formula into Formula and using the Substitution Rule, we obtain t b L d d a d ddt d ddt dt dt P P C P i _ P i Since ddt, we have L dt d dt d dt FIGURE P P n Even if C can t be epressed in the form F, Formula is still valid but we obtain it b polgonal approimations. We divide the parameter interval, into n subintervals of equal width t. If t, t, t,..., t n are the endpoints of these subintervals, then i f t i and i tt i are the coordinates of points P i i, i that lie on C and the polgon with vertices P, P,..., P n approimates C. (See Figure.) As in Section 8., we define the length L of C to be the limit of the lengths of these approimating polgons as nl: L lim nl n i P ip i The Mean Value Theorem, when applied to f on the interval t i, t i, gives a number t i * in t i, t i such that f t i f t i f t i *t i t i If we let i i i and i i i, this equation becomes i f t i * t

15 6 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES Similarl, when applied to t, the Mean Value Theorem gives a number t i ** in t i, t i such that i tt i ** t Therefore P ip i s i i s f t i * tt i ** t and so s f t i *t tt i **t 5 L lim nl n s f t i * tt i ** t i The sum in (5) resembles a Riemann sum for the function s f t tt but it is not eactl a Riemann sum because t i * t i ** in general. Nevertheless, if f and t are continuous, it can be shown that the limit in (5) is the same as if t i * and t i ** were equal, namel, Thus, using Leibniz notation, we have the following result, which has the same form as Formula (). 6 THEREM If a curve C is described b the parametric equations f t, tt,, where f and t are continuous on, and C is traversed eactl once as t increases from to, then the length of C is L dt dt d dt t L d s f t tt dt Notice that the formula in Theorem 6 is consistent with the general formulas L ds and ds d d of Section 8.. EXAMPLE If we use the representation of the unit circle given in Eample in Section., cos t sin t t then ddt sin t and ddt cos t, so Theorem 6 gives L dt d dt d dt ssin t cos t dt as epected. If, on the other hand, we use the representation given in Eample in Section., sin t cos t t then ddt cos t, ddt sin t, and the integral in Theorem 6 gives dt d dt d dt s cos t sin t dt dt dt

16 SECTIN. CALCULUS WITH PARAMETRIC CURVES 65 Notice that the integral gives twice the arc length of the circle because as t increases from to, the point sin t, cos t traverses the circle twice. In general, when finding the length of a curve C from a parametric representation, we have to be careful to ensure that C is traversed onl once as t increases from to. M V EXAMPLE 5 Find the length of one arch of the ccloid r sin, r cos. SLUTIN From Eample we see that one arch is described b the parameter interval. Since d d r cos and d d r sin N The result of Eample 5 sas that the length of one arch of a ccloid is eight times the radius of the generating circle (see Figure 5). This was first proved in 658 b Sir Christopher Wren, who later became the architect of St. Paul s Cathedral in London. r L=8r we have L d d d d d sr cos cos sin d r s cos d To evaluate this integral we use the identit sin cos with, which gives cos sin. Since, we have and so sin. Therefore s cos s sin sin sin sr cos r sin d πr and so L r sin d r cos] FIGURE 5 r 8r M SURFACE AREA In the same wa as for arc length, we can adapt Formula 8..5 to obtain a formula for surface area. If the curve given b the parametric equations f t, tt,, is rotated about the -ais, where f, t are continuous and tt, then the area of the resulting surface is given b 7 S The general smbolic formulas S ds and S ds (Formulas 8..7 and 8..8) are still valid, but for parametric curves we use ds dt d dt d dt EXAMPLE 6 Show that the surface area of a sphere of radius r is r. SLUTIN The sphere is obtained b rotating the semicircle r cos t dt d dt d dt r sin t t t

17 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES about the -ais. Therefore, from Formula 7, we get S r sin t sr sin t r cos t dt r r sin t sr sin t cos t dt sin t dt r cos t] r r sin t r dt M. EXERCISES Find dd.. t sin t, t t. t, st e t 9. cos,. cos, sin sin 6 Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.. t, t t;. t t, t ; 5. e st, t ln t ; 6. cos sin, sin cos ; 7 8 Find an equation of the tangent to the curve at the given point b two methods: (a) without eliminating the parameter and (b) b first eliminating the parameter. ; 9 Find an equation of the tangent(s) to the curve at the given point. Then graph the curve and the tangent(s) sin t, t t;. cos t cos t, sin t sin t ; 6 Find dd and d d. For which values of t is the curve concave upward?. t, t t. t t, t. t e t, t e t. t ln t, t ln t 5. sin t, cos t, 6. cos t, cos t, 7 Find the points on the curve where the tangent is horizontal or vertical. If ou have a graphing device, graph the curve to check our work. 7. t, t t t t t 7. ln t, t ;, 8. tan, sec ; (, s), t t 8. t t t, t t, ;. Use a graph to estimate the coordinates of the rightmost point on the curve t t 6, e t. Then use calculus to find the eact coordinates. ;. Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve t t, t t. Then find the eact coordinates. ; Graph the curve in a viewing rectangle that displas all the important aspects of the curve.. t t t,. t t 8t, 5. Show that the curve cos t, sin t cos t has two tangents at, and find their equations. Sketch the curve. ; 6. Graph the curve cos t cos t, sin t sin t to discover where it crosses itself. Then find equations of both tangents at that point. 7. (a) Find the slope of the tangent line to the trochoid r d sin, r d cos in terms of. (See Eercise in Section..) (b) Show that if d r, then the trochoid does not have a vertical tangent. 8. (a) Find the slope of the tangent to the astroid, a sin a cos in terms of. (Astroids are eplored in the Laborator Project on page 69.) (b) At what points is the tangent horizontal or vertical? (c) At what points does the tangent have slope or? 9. At what points on the curve t, t t does the tangent line have slope?. Find equations of the tangents to the curve t, t that pass through the point,.. Use the parametric equations of an ellipse, a cos, b sin,, to find the area that it encloses. t t t t

18 SECTIN. CALCULUS WITH PARAMETRIC CURVES 67. Find the area enclosed b the curve t t, st and the -ais.. Find the area enclosed b the -ais and the curve e t, t t.. Find the area of the region enclosed b the astroid a cos, a sin. (Astroids are eplored in the Laborator Project on page 69.) a _a a 9. Use Simpson s Rule with n 6 to estimate the length of the curve t e t, t e t, 6 t In Eercise in Section. ou were asked to derive the parametric equations a cot, a sin for the curve called the witch of Maria Agnesi. Use Simpson s Rule with n to estimate the length of the arc of this curve given b. 5 5 Find the distance traveled b a particle with position, as t varies in the given time interval. Compare with the length of the curve. 5. sin t, cos t, t 5. cos t, cos t, t 5. Find the area under one arch of the trochoid of Eercise in Section. for the case d r. 6. Let be the region enclosed b the loop of the curve in Eample. (a) Find the area of. (b) If is rotated about the -ais, find the volume of the resulting solid. (c) Find the centroid of. 7 Set up an integral that represents the length of the curve. Then use our calculator to find the length correct to four decimal places. 7. t t, t, t 8. e t, t, t 9. t cos t, t sin t,. ln t, st, Find the eact length of the curve.. t, t, t. e t e t, 5 t, t. t, ln t, t. cos t cos t, sin t sin t, ; 5 7 Graph the curve and find its length. 5. e t cos t, e t sin t, t 6. cos t ln(tan t), sin t, t 7. e t t, e t, _a t 5 t 8 t t t 8. Find the length of the loop of the curve t t, t. CAS CAS 5. Show that the total length of the ellipse a sin, b cos, a b, is L a where e is the eccentricit of the ellipse (e ca, where c sa b ). 5. Find the total length of the astroid a cos, a sin, where a. 55. (a) Graph the epitrochoid with equations cos t cost sin t sint What parameter interval gives the complete curve? (b) Use our CAS to find the approimate length of this curve. 56. A curve called Cornu s spiral is defined b the parametric equations Ct t St t where C and S are the Fresnel functions that were introduced in Chapter 5. (a) Graph this curve. What happens as tl and as tl? (b) Find the length of Cornu s spiral from the origin to the point with parameter value t Set up an integral that represents the area of the surface obtained b rotating the given curve about the -ais. Then use our calculator to find the surface area correct to four decimal places. 57. te t, t e t, cosu du sinu du t 58. sin t, sin t, t s e sin d

19 68 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES 59 6 Find the eact area of the surface obtained b rotating the given curve about the -ais. 59. t, t, t 6. t t, t, t 6. a cos, a sin, ; 6. Graph the curve If this curve is rotated about the -ais, find the area of the resulting surface. (Use our graph to help find the correct parameter interval.) 6. If the curve cos cos is rotated about the -ais, use our calculator to estimate the area of the resulting surface to three decimal places. 6. If the arc of the curve in Eercise 5 is rotated about the -ais, estimate the area of the resulting surface using Simpson s Rule with n Find the surface area generated b rotating the given curve about the -ais. 66. e t t, e t, 67. If f is continuous and f t for a t b, show that the parametric curve f t, tt, a t b, can be put in the form F. [Hint: Show that f eists.] 68. Use Formula to derive Formula 7 from Formula 8..5 for the case in which the curve can be represented in the form F, a b. 69. The curvature at a point P of a curve is defined as ds t t 65. t, t, t t t 5 t d where is the angle of inclination of the tangent line at P, as shown in the figure. Thus the curvature is the absolute value of the rate of change of with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at P and will be studied in greater detail in Chapter. (a) For a parametric curve t, t, derive the formula sin sin t (b) B regarding a curve f as the parametric curve, f, with parameter, show that the formula in part (a) becomes 7. (a) Use the formula in Eercise 69(b) to find the curvature of the parabola at the point,. (b) At what point does this parabola have maimum curvature? 7. Use the formula in Eercise 69(a) to find the curvature of the ccloid, cos at the top of one of its arches. 7. (a) Show that the curvature at each point of a straight line is. (b) Show that the curvature at each point of a circle of radius r is. sin 7. A string is wound around a circle and then unwound while being held taut. The curve traced b the point P at the end of the string is called the involute of the circle. If the circle has radius r and center and the initial position of P is r,, and if the parameter is chosen as in the figure, show that parametric equations of the involute are rcos d d r sin dd 7. A cow is tied to a silo with radius r b a rope just long enough to reach the opposite side of the silo. Find the area available for grazing b the cow. r T rsin P P cos where the dots indicate derivatives with respect to t, so ddt. [Hint: Use and Formula to find ddt. Then use the Chain Rule to find dds.] tan dd

20 SECTIN. PLAR CRDINATES 69 L A B R AT R Y P R J E C T ; BÉZIER CURVES The Bézier curves are used in computer-aided design and are named after the French mathematician Pierre Bézier (9 999), who worked in the automotive industr. A cubic Bézier curve is determined b four control points, P,, P,, P,, and P,, and is defined b the parametric equations t t t t t t t t t t t t where t. Notice that when t we have,, and when t we have,,, so the curve starts at and ends at.. Graph the Bézier curve with control points P,, P 8, 8, P 5,, and P, 5. Then, on the same screen, graph the line segments P P, P P, and P P. (Eercise in Section. shows how to do this.) Notice that the middle control points P and P don t lie on the curve; the curve starts at P, heads toward P and P without reaching them, and ends at P.. From the graph in Problem, it appears that the tangent at P passes through P and the tangent at P passes through. Prove it. P P. Tr to produce a Bézier curve with a loop b changing the second control point in Problem.. Some laser printers use Bézier curves to represent letters and other smbols. Eperiment with control points until ou find a Bézier curve that gives a reasonable representation of the letter C. 5. More complicated shapes can be represented b piecing together two or more Bézier curves. Suppose the first Bézier curve has control points P, P, P, P and the second one has control points P, P, P 5, P 6. If we want these two pieces to join together smoothl, then the tangents at P should match and so the points P, P, and P all have to lie on this common tangent line. Using this principle, find control points for a pair of Bézier curves that represent the letter S. P. PLAR CRDINATES FIGURE r polar ais P(r, ) A coordinate sstem represents a point in the plane b an ordered pair of numbers called coordinates. Usuall we use Cartesian coordinates, which are directed distances from two perpendicular aes. Here we describe a coordinate sstem introduced b Newton, called the polar coordinate sstem, which is more convenient for man purposes. We choose a point in the plane that is called the pole (or origin) and is labeled. Then we draw a ra (half-line) starting at called the polar ais. This ais is usuall drawn horizontall to the right and corresponds to the positive -ais in Cartesian coordinates. If P is an other point in the plane, let r be the distance from to P and let be the angle (usuall measured in radians) between the polar ais and the line P as in Figure. Then the point P is represented b the ordered pair r, and r, are called polar coordinates of P. We use the convention that an angle is positive if measured in the counterclockwise direction from the polar ais and negative in the clockwise direction. If P, then r and we agree that, represents the pole for an value of.

21 6 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES +π (r, ) We etend the meaning of polar coordinates r, to the case in which r is negative b agreeing that, as in Figure, the points r, and r, lie on the same line through and at the same distance r from, but on opposite sides of. If r, the point r, lies in the same quadrant as ; if r, it lies in the quadrant on the opposite side of the pole. Notice that r, represents the same point as r,. (_r, ) FIGURE 5π 5π, (, π) EXAMPLE Plot the points whose polar coordinates are given. (a), 5 (b), (c), (d), SLUTIN The points are plotted in Figure. In part (d) the point, is located three units from the pole in the fourth quadrant because the angle is in the second quadrant and r is negative. π π _ π FIGURE π, _ π _, M In the Cartesian coordinate sstem ever point has onl one representation, but in the polar coordinate sstem each point has man representations. For instance, the point, 5 in Eample (a) could be written as, or, or,. (See Figure.) 5π _ π π π 5π, π, _ π, π _, FIGURE In fact, since a complete counterclockwise rotation is given b an angle, the point represented b polar coordinates r, is also represented b r, n and r, n r P(r, )=P(, ) where n is an integer. The connection between polar and Cartesian coordinates can be seen from Figure 5, in which the pole corresponds to the origin and the polar ais coincides with the positive -ais. If the point P has Cartesian coordinates, and polar coordinates r,, then, from the figure, we have cos sin r r and so FIGURE 5 r cos r sin Although Equations were deduced from Figure 5, which illustrates the case where r and, these equations are valid for all values of r and. (See the general definition of sin and cos in Appendi D.)

22 SECTIN. PLAR CRDINATES 6 Equations allow us to find the Cartesian coordinates of a point when the polar coordinates are known. To find r and when and are known, we use the equations r tan which can be deduced from Equations or simpl read from Figure 5. EXAMPLE Convert the point, from polar to Cartesian coordinates. SLUTIN Since r and, Equations give r cos cos r sin sin Therefore the point is (, s ) in Cartesian coordinates. M EXAMPLE Represent the point with Cartesian coordinates, in terms of polar coordinates. SLUTIN If we choose r to be positive, then Equations give s s r s s s tan Since the point, lies in the fourth quadrant, we can choose or. Thus one possible answer is (s, ) ; another is s, 7. M 7 NTE Equations do not uniquel determine when and are given because, as increases through the interval, each value of tan occurs twice. Therefore, in converting from Cartesian to polar coordinates, it s not good enough just to find r and that satisf Equations. As in Eample, we must choose so that the point r, lies in the correct quadrant. r= r= r= r= PLAR CURVES The graph of a polar equation r f, or more generall Fr,, consists of all points P that have at least one polar representation r, whose coordinates satisf the equation. V EXAMPLE What curve is represented b the polar equation r? FIGURE 6 SLUTIN The curve consists of all points r, with r. Since r represents the distance from the point to the pole, the curve r represents the circle with center and radius. In general, the equation r a represents a circle with center and radius a. (See Figure 6.) M

23 6 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES = (_, ) (_, ) (, ) (, ) (, ) EXAMPLE 5 Sketch the polar curve. SLUTIN This curve consists of all points r, such that the polar angle is radian. It is the straight line that passes through and makes an angle of radian with the polar ais (see Figure 7). Notice that the points r, on the line with r are in the first quadrant, whereas those with r are in the third quadrant. M EXAMPLE 6 (a) Sketch the curve with polar equation r cos. (b) Find a Cartesian equation for this curve. FIGURE 7 SLUTIN (a) In Figure 8 we find the values of r for some convenient values of and plot the corresponding points r,. Then we join these points to sketch the curve, which appears to be a circle. We have used onl values of between and, since if we let increase beond, we obtain the same points again. FIGURE 8 Table of values and graph of r= cos 6 s s s 56 s r cos π, π, π _, π œ, π œ, 6 (, ) 5π _ œ, π 6 _ œ, (b) To convert the given equation to a Cartesian equation we use Equations and. From r cos we have cos r, so the equation r cos becomes r r, which gives r or Completing the square, we obtain which is an equation of a circle with center, and radius. M N Figure 9 shows a geometrical illustration that the circle in Eample 6 has the equation r cos. The angle PQ is a right angle (Wh?) and so r cos. r P Q FIGURE 9

24 SECTIN. PLAR CRDINATES 6 r π π π π FIGURE r=+sin in Cartesian coordinates, π V EXAMPLE 7 Sketch the curve r sin. SLUTIN Instead of plotting points as in Eample 6, we first sketch the graph of r sin in Cartesian coordinates in Figure b shifting the sine curve up one unit. This enables us to read at a glance the values of r that correspond to increasing values of. For instance, we see that as increases from to, r (the distance from ) increases from to, so we sketch the corresponding part of the polar curve in Figure (a). As increases from to, Figure shows that r decreases from to, so we sketch the net part of the curve as in Figure (b). As increases from to, r decreases from to as shown in part (c). Finall, as increases from to, r increases from to as shown in part (d). If we let increase beond or decrease beond, we would simpl retrace our path. Putting together the parts of the curve from Figure (a) (d), we sketch the complete curve in part (e). It is called a cardioid, because it s shaped like a heart. = π = π = =π =π =π = π = π (a) (b) (c) (d) (e) FIGURE Stages in sketching the cardioid r=+sin M TEC Module. helps ou see how polar curves are traced out b showing animations similar to Figures. EXAMPLE 8 Sketch the curve r cos. SLUTIN As in Eample 7, we first sketch r cos,, in Cartesian coordinates in Figure. As increases from to, Figure shows that r decreases from to and so we draw the corresponding portion of the polar curve in Figure (indicated b!). As increases from to, r goes from to. This means that the distance from increases from to, but instead of being in the first quadrant this portion of the polar curve (indicated lies on the opposite side of the pole in the third quadrant. The remainder of the curve is drawn in a similar fashion, with the arrows and numbers indicating the order in which the portions are traced out. The resulting curve has four loops and is called a four-leaved rose. r = π! $ % * = π $ & ^! = π π π π # ^ & 5π π 7π π =π # 8 = FIGURE r=cos in Cartesian coordinates FIGURE Four-leaved rose r=cos M

25 6 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SYMMETRY When we sketch polar curves, it is sometimes helpful to take advantage of smmetr. The following three rules are eplained b Figure. (a) If a polar equation is unchanged when is replaced b, the curve is smmetric about the polar ais. (b) If the equation is unchanged when r is replaced b r, or when is replaced b, the curve is smmetric about the pole. (This means that the curve remains unchanged if we rotate it through 8 about the origin.) (c) If the equation is unchanged when is replaced b, the curve is smmetric about the vertical line. (r, ) (r, π- ) (r, ) _ (_r, ) (r, ) π- (r, _ ) (a) (b) (c) FIGURE The curves sketched in Eamples 6 and 8 are smmetric about the polar ais, since cos cos. The curves in Eamples 7 and 8 are smmetric about because sin and cos cos. The four-leaved rose is also smmetric about the pole. These smmetr properties could have been used in sketching the curves. For instance, in Eample 6 we need onl have plotted points for and then reflected about the polar ais to obtain the complete circle. sin TANGENTS T PLAR CURVES To find a tangent line to a polar curve r f, we regard as a parameter and write its parametric equations as r cos f cos r sin f sin Then, using the method for finding slopes of parametric curves (Equation..) and the Product Rule, we have We locate horizontal tangents b finding the points where (provided that ). Likewise, we locate vertical tangents at the points where (provided that ). Notice that if we are looking for tangent lines at the pole, then r and Equation simplifies to d dr if d tan dd dd d d d d d d dr d dr d sin r cos cos r sin d dd dd

26 SECTIN. PLAR CRDINATES 65 For instance, in Eample 8 we found that r cos when or. This means that the lines and (or and ) are tangent lines to r cos at the origin. EXAMPLE 9 (a) For the cardioid r sin of Eample 7, find the slope of the tangent line when. (b) Find the points on the cardioid where the tangent line is horizontal or vertical. SLUTIN Using Equation with r sin, we have d d (a) The slope of the tangent at the point where is d d cos sin sin sin dr d dr d sin r cos cos r sin cos sin sin cos sin sin sin sin cos sin sin cos cos cos sin sin ( s ) ( s )( s ) (b) bserve that s ( s )( s ) s s d d cos sin when,, 7 6, 6 d d sin sin when, 6, 5 6 π, m=_ œ π +, Therefore there are horizontal tangents at the points,, (, 76), (, 6) and vertical tangents at (, 6) and (, 56). When, both and are, so we must be careful. Using l Hospital s Rule, we have d lim l d sin cos lim lim l sin l sin dd dd, 5π 6 (, ) π, 6 cos lim l sin sin lim l cos 7π π, 6, 6 FIGURE 5 Tangent lines for r=+sin B smmetr, d lim l d Thus there is a vertical tangent line at the pole (see Figure 5). M

27 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES NTE Instead of having to remember Equation, we could emplo the method used to derive it. For instance, in Eample 9 we could have written Then we would have d d r cos sin cos cos sin r sin sin sin sin sin dd dd which is equivalent to our previous epression. cos sin cos cos sin sin cos sin cos GRAPHING PLAR CURVES WITH GRAPHING DEVICES Although it s useful to be able to sketch simple polar curves b hand, we need to use a graphing calculator or computer when we are faced with a curve as complicated as the ones shown in Figures 6 and _ FIGURE 6 r=sin@(. )+cos$(. ) _.7 FIGURE 7 r=sin@(. )+cos#(6 ) Some graphing devices have commands that enable us to graph polar curves directl. With other machines we need to convert to parametric equations first. In this case we take the polar equation r f and write its parametric equations as r cos f cos Some machines require that the parameter be called t rather than. EXAMPLE Graph the curve r sin85. r sin f sin SLUTIN Let s assume that our graphing device doesn t have a built-in polar graphing command. In this case we need to work with the corresponding parametric equations, which are r cos sin85 cos r sin sin85 sin In an case, we need to determine the domain for. So we ask ourselves: How man complete rotations are required until the curve starts to repeat itself? If the answer is n, then sin 8 n 5 sin 8 5 6n sin 8 5 5