10.4 AREAS AND LENGTHS IN POLAR COORDINATES


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1 65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the formul for the re of sector of circle r A r FIGURE = 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 Exercise 35 in Section 7.3.) Let e the region, illustrted in Figure, ounded y the polr curve r f nd y the rys nd, where f is positive continuous function nd where. We divide the intervl, into suintervls with endpoints,,,..., n nd equl width. The rys i then divide into n smller regions with centrl ngle i i. If we choose i* in the ith suintervl i, i, then the re A i of the ith region is pproximted y 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* = FIGURE 3 f( i*) = = i = i Î nd so n pproximtion to the totl re A of is It ppers from Figure 3 tht the pproximtion in () improves s n l. But the sums in () re Riemnn sums for the function t f,so lim n l n i A n i f i* f i* y f d It therefore ppers plusile (nd cn in fct e proved) tht the formul for the re A of the polr region is 3 A y f d Formul 3 is often written s 4 A y r d with the understnding tht r f. Note the similrity etween Formuls nd 4. When we pply Formul 3 or 4, it is helpful to think of the re s eing swept out y rotting ry through tht strts with ngle nd ends with ngle. V EXAMPLE Find the re enclosed y one loop of the fourleved rose r cos. SLUTIN The curve r cos ws sketched in Exmple 8 in Section.3. Notice from Figure 4 tht the region enclosed y the right loop is swept out y ry tht rottes from
2 SECTIN.4 AREAS AND LENGTHS IN PLAR CRDINATES 65 r=cos = π 4 4 to 4 4 A y. Therefore Formul 4 gives 4 r d 4 y y 4 cos d 4 cos d y 4 [ cos 4 d 4 sin 4] 4 8 M =_ π 4 V EXAMPLE Find the re of the region tht lies inside the circle r 3 sin nd outside the crdioid r sin. FIGURE 4 r=3 sin SLUTIN The crdioid (see Exmple 7 in Section.3) nd the circle re sketched in Figure 5 nd the desired region is shded. The vlues of nd in Formul 4 re determined y finding the points of intersection of the two curves. They intersect when 3 sin sin,which gives sin,so, 56. The desired re cn e found y sutrcting the re inside the crdioid etween nd from the re inside the circle from 6 to 56. Thus = 5π 6 = π 6 A y 56 3sin d y 56 sin d 6 6 FIGURE 5 r=+sin Since the region is symmetric out the verticl xis, we cn write A y 9 sin d y sin sin d 6 6 y 6 8 sin sin d y cos sin d [ecuse sin cos ] 3 sin cos ]6 M = r=g( ) = FIGURE 6 r=f( ) Exmple illustrtes the procedure for finding the re of the region ounded y two polr curves. In generl, let e region, s illustrted in Figure 6, tht is ounded y curves with polr equtions r f, r t,, nd, where f t nd. The re A of is found y sutrcting the re inside r t from the re inside r f, so using Formul 3 we hve A y y f t f d y t d d CAUTIN The fct tht single point hs mny representtions in polr coordintes sometimes mkes it difficult to find ll the points of intersection of two polr curves. For instnce, it is ovious from Figure 5 tht the circle nd the crdioid hve three points of intersection; however, in Exmple we solved the equtions r 3sin nd r sin nd found only two such points, ( 3, 6) nd ( 3, 56). The origin is lso point of intersection, ut we cn t find it y solving the equtions of the curves ecuse the origin hs no single representtion in polr coordintes tht stisfies oth equtions. Notice tht, when represented s, or,, the origin stisfies r 3 sin 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
3 65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES r= FIGURE 7 π, 3 π, 6 r=cos other curve it is reched t. The points don t collide t the origin ecuse they rech the origin t different times, ut the curves intersect there nonetheless. Thus, to find ll points of intersection of two polr curves, it is recommended tht you drw the grphs of oth curves. It is especilly 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, 5 3, 73, 3. Thus the vlues of etween nd tht stisfy oth equtions re 6, 5 6, 76, 6. We hve found four points of intersection: (, 6), (, 56), (, 76),nd(, 6). However, you cn see from Figure 7 tht the curves hve four other points of intersection nmely, (, 3), (, 3), (, 43), nd (, 53). These cn e found using symmetry or y noticing tht nother eqution of the circle is r nd then solving the equtions r cos nd r. M ARC LENGTH To find the length of polr curve r f,, we regrd s prmeter nd write the prmetric equtions of the curve s Using the Product Rule nd differentiting with respect to, we otin so, using cos sin, we hve dx dr cos r dr Assuming tht f is continuous, we cn use Theorem..6 to write the rc length s L y dx Therefore the length of curve with polr eqution r f,, is 5 dx dr cos r sin d d dy x r cos f cos d d 3 d dr dr r d L y d d d sin r dr d dy r dr y r sin f sin dy dr sin r cos d d d cos sin r sin d d d sin cos r cos V EXAMPLE 4 Find the length of the crdioid r sin. SLUTIN The crdioid is shown in Figure 8. (We sketched it in Exmple 7 in Section.3.) Its full length is given y the prmeter intervl,so
4 SECTIN.4 AREAS AND LENGTHS IN PLAR CRDINATES 653 FIGURE 8 r=+sin Formul 5 gives L y r dr y s sin d d d y We could evlute this integrl y multiplying nd dividing the integrnd y s sin,or we could use computer lger system. In ny event,we find tht the length of the crdioid is L 8. M s sin cos d.4 EXERCISES 4 Find the re of the region tht is ounded y the given curve nd lies in the specified sector.. r, 4. r e, 3. r sin, 3 4. r ssin, 5 8 Find the re of the shded region r=œ 9 4 Sketch the curve nd find the re tht it encloses.. r 4 cos r=4+3 sin r=+cos r=sin 9. r 3 cos. r 3 cos r sin 3. r cos 3 4. r cos ; 5 6 Grph the curve nd find the re tht it encloses. 7 Find the re of the region enclosed y one loop of the curve. 7. r sin 8. r 4 sin 3 5. r sin 6 6. r sin 3 sin 9 9. r 3 cos 5. r sin 6. r sin (inner loop). Find the re enclosed y 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, r 6. r sin, 7. r 3cos, 8. r 3sin, 9 34 Find the re of the region tht lies inside oth curves. 9. r s3 cos, 3. r cos, 3. r sin, 3. r 3 cos, 33. r sin, 34. r sin, r cos,, 35. Find the re inside the lrger loop nd outside the smller loop of the limçon r cos. 36. Find the re etween 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, r cos r sin r sin r cos r cos r 3 sin r cos r 3 sin r sin 39. r sin, r 4. r cos 3, r r 3sin r sin 3 4. r sin, r sin 4. r sin, r cos
5 654 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES ; 43. The points of intersection of the crdioid r sin nd the spirl loop r,,cn t e found exctly. Use grphing device to find the pproximte vlues of t which they intersect. Then use these vlues to estimte the re tht lies inside oth curves. 44. When recording live performnces, sound engineers often use microphone with crdioid pickup pttern ecuse 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 oundry of the optiml pickup region is given y 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 they will hve on stge within the optiml pickup rnge of the microphone. Answer their question Find the exct length of the polr curve. 45. r 3 sin, r e, 47. r stge udience m 4 m microphone, 48. r, 49 5 Use clcultor to find the length of the curve correct to four deciml plces. 49. r 3 sin 5. r 4 sin 3 5. r sin 5. r cos3 ; Grph the curve nd find its length. 53. r cos r cos 55. () Use Formul..7 to show tht the re of the surfce generted y rotting the polr curve (where f is continuous nd ) out the polr xis is S y r f r sin r d dr () Use the formul in prt () to find the surfce re generted y rotting the lemniscte r cos out the polr xis. 56. () Find formul for the re of the surfce generted y rotting the polr curve r f, (where f is continuous nd ), out the line. () Find the surfce re generted y rotting the lemniscte r cos out the line. d.5 CNIC SECTINS In this section we give geometric definitions of prols, ellipses, nd hyperols nd derive their stndrd equtions. They re clled conic sections, or conics, ecuse they result from intersecting cone with plne s shown in Figure. ellipse prol hyperol FIGURE Conics
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