Area and Arc Length in Polar CoordinatesArea of a Sector of a CircleArea of a Sector of a RegionSlide 4GuidelinesFind the AreaAreas of Portions of a RegionArea of a Single LoopIntersections of Polar EquationsStrategiesA Sneaky ProblemHintsArea Of IntersectionArc LengthTry it Out!AssignmentArea and Arc Length in Polar CoordinatesLesson 10.5Area of a Sector of a Circle•Given a circle with radius = rSector of the circle with angle = θ•The area of the sector given by θr212A rq=Area of a Sector of a Region•Consider a region bounded by r = f(θ)•A small portion (a sector with angle dθ) has areadθα••β[ ]21( )2A f dq q�Area of a Sector of a Region•We use an integral to sum the small pie slicesα••β[ ]221( )212A f dr dbabaq qq==��r = f(θ)Guidelines1. Use the calculator to graph the region•Find smallest value θ = a, and largest value θ = b for the points (r, θ) in the region2. Sketch a typical circular sector•Label central angle dθ3. Express the area of the sector as4. Integrate the expression over the limits from a to b212A rq=Find the Area•Given r = 4 + sin θFind the area of the region enclosed by the ellipsedθ( )22014 sin2dpq q+�The ellipse is traced out by 0 < θ < 2πThe ellipse is traced out by 0 < θ < 2πAreas of Portions of a Region•Given r = 4 sin θ and rays θ = 0, θ = π/3/ 320116sin2dpq q�The angle of the rays specifies the limits of the integrationThe angle of the rays specifies the limits of the integrationArea of a Single Loop•Consider r = sin 6θNote 12 petalsθ goes from 0 to 2πOne loop goes from0 to π/6( )/ 6201sin 62dpq q�Intersections of Polar Equations•To find area of intersecting regions•Need to know where the graphs intersect9• r = 1• r = 2 cos θ• r = 1• r = 2 cos θStrategies•Use substitutionLet r = 1 in the second equationSolve for θLet @n1 = 0, result is10• r = 1• r = 2 cos θ• r = 1• r = 2 cos θ3 3andp p-A Sneaky Problem•Consider r = sin θ and r = cos θ•What is simultaneoussolution?Where sin θ = cos θ that is•Problem … the intersection at the pole does not show up using this strategyYou must inspect the graph112,2 4p� �� �� �� �Hints1. Graph the curves on your calculatora) Observe the number of intersectionsb) Zoom in as needed2. Do a simultaneous solution to the two equationsa) Check results against observed points of intersectionb) Discard duplicatesc) Note intersection at the pole that simultaneous solutions may not have given12Area Of Intersection•Note the area that is inside r = 2 sin θand outside r = 1•Find intersections•Consider sector for a dθMust subtract two sectorsdθ56 6andp p56 6andp p( )5 / 622/ 612sin 12dppq q� �-� ��14Arc Length•Given a curve in polar form r = f (θ)Must have continuous first derivative on intervalCurve must be traced exactly once for a ≤ θ ≤ b•Arc length is( ) ( )2 222'babaL f f ddrr ddq q qqq= +� � � �� � � �� �= +� �� ���15Try it Out!•Given polar functionWhat is the arc length from θ = 0 to θ = 4•Find dr/dθ•What is the integral and its evaluation/ 2r eq=/ 21'2r eq= �( )44 4/ 2 / 2 20 001 5 52 1 54 2 2e e d e d e eq q q qq q+ = = � = -� �Assignment•Lesson 10.5•Page 745•Exercises 1 – 37
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