ExercisesSlopes of Polr CurvesFind the slopes of the curves in Exercises 17-20 at the given points17.-1,117, CardioidF = + cos0; 0=±n/219, Four-leaved rose (r = sin 28; o = ±#/4, ±3#/419.-1,1,1, -111.5Areasand LengthsinPolar CoordinatesSlide 3-31Slide 3-32Theregion OTS inFigure 1130isbounded by theryseando βanmecureFflo),Weapproximatetheregion withnonoverlapping fan-shaped circularsectors based on a partition Pof angle TOS, The typical sector has radius n = (o,) andcentral angle of radian measure Aoy, Its area is e/2rr times the area of a circle ofradius rorF-8(r(oho)1.AreainthePlaneThe area ofregion OrS is approximately4-21(r0) s0.-fePPU,9FIGURE1.31The area differcatiald.4Slide 3- 33Slide 3- 34for thecurver = fo)EXAMPLE1Find the area of the region in the plane enclosed by the cardoidI /iseomtinuous, we expet the approximatios to improseasthe om othe parti= 2(1. + cos.0),tionPgoestozero,where thenormofPithelargestvalucofe,Weare then led totSolutionWe graph thecardioid (Figure 11.32) and determine that theradius OP sweepsfollowing formula defining the regionls arexout the region exactly once as 9 runs from O to 2,The area is thereforeicrom4-limC-o201+cm8["1(r0)'aP)2(1+ 2cos0 + 0P 0) d0n,2g口(2+400+21+pm2) aArea of the Fan-ShRegion Between the Origin and the CurvF-reasesg(3+ 4s+cos20)do["lraA-[30 +4i0 +]Thisisteinegalftheareadiffereial (Figure131) 6# 0.= 6#d(r(o)amSlide 3- 35slide 3-38
2016/11/15 6 Slide 3 - 31 Exercises 17. -1, 1 19. -1, 1, 1, -1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 32 11.5 Areas and Lengths in Polar Coordinates Slide 3 - 33 1. Area in the Plane Slide 3 - 34 Slide 3 - 35 Slide 3 - 36
Find theaoeregiothatlisinsidete cireled outsideEXAMPLE2cardioidr=1-cose.the area ofthe regionwhichWesketch the regionto determine itsboundaries and find the limits of integraSolutlonliesbetweentwopolartion (Figure 11.34), The outer curve is r2= 1, the inner curve is n =1cos e, and curves r,=r,(e) and r2-r(e)runs from -/2to w/2.The area, from Equation (1),isfromg=atoe=β1(rr2)do44P(c2-)d0SymmeiryFIGURE11.33The areaofthe shadedregion is.calculated by subtracting the arcs(1 (1 2cos8 + cos 0) dooftheregian botwen n und iiecoriginSqanrenfrom theareaofthergicnbetwemandthe ocipin(2 0os 0 coe2 0) dLomarlini20s6 L+gm20)Area of the Reglou 05rifo)5SnasosAld-''do-'()ao(b)8in 20 /r/28ino-号=2-弄Slide 3-37Slide 3-38Finding Polar AreasFind the areas of the rogions in Exercises 1-8,4, Inside the cardioid r = a(1 + cos.e), a > 0ExercisesExercises1. Boended by the spinl r = e for o s 8 s 5. Inside one leaf ofthe four-lened rose r = cos 206, Inside one leaf of the three-leaved rose r m cos 38(6)rCm3e(n, w)2, Bounded by the cireler = 2 sin 9 for /4 S 9≤ /2动ide 3- 39Slide 3- 4014, Insido the circle r 3a cos and outside the cardioidr=a(1 +00s8),a>ExercisesExercises15, lsside the circle r = 2oos @ ad outside the circler =Find the areas of the regions in Exercises 9-1616, laside the circle r = 6 sbove the tiner 3esee9, Shared by the circles r.= 2 cos and r = 2 sin g17, tnside the circler= 4coseand to tho rightofthe vertical linc10, Shareod by the circles r = 1 and r. 2 sin 8r=secd11. Shared by the circle r = 2 and the candioid r = 2(1 cos e)18. Inside the circle r= 4 sin & and bekow the horisontal lincr-3esc812. Shared by the cardioids r = 2(1 + cos 0) and r = 2(1 cos 0)19, a, Find the area of thparyitg figure13, Inside the lemmiscater = 6 cos 28 and outide the circler =. V-OC9.(2snordo+（200=/4)2sin 0de+2cos* 0d0=I(1cos20)do+J(1+cos 20)d0V2/20c--{0-2-0+2--0h. ltlooks as ifthe graph of r =tan g, e/2<0<w/2,could be asymplotic tothe linesx 1andx 1.is it?Give nesons for your answer.Slide 3-41slide3-42
2016/11/15 7 Slide 3 - 37 the area of the region which lies between two polar curves r1=r1 (θ) and r2=r2 (θ) from θ=α to θ=β Slide 3 - 38 Slide 3 - 39 Exercises Slide 3 - 40 Exercises Slide 3 - 41 Exercises /4 /2 2 2 0 /4 /4 /2 /4 /2 2 2 0 /4 0 /4 /4 /2 0 /4 1 1 9. (2sin ) (2cos ) 2 2 2sin 2cos (1 cos 2 ) (1 cos 2 ) 1 1 sin 2 sin 2 1 2 2 2 d d d d d d Slide 3 - 42 Exercises