11.3 Polar Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall What you’ll learn about… …and why n Polar Coordinates Polar equations enable us to n Polar Curves define some interesting and n Slopes of Polar Curves important curves that would n Areas Enclosed by Polar Curves be difficult or impossible to n A Small Polar Gallery define in the form y = f(x). Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 2 Polar-Rectangular Conversion Formulas 2 2 2 x = r cosθ r = x + y y y = r sinθ tanθ = x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 3 Example Rectangular and Polar Coordinates Find rectangular coordinates for the point with the polar coordinate: ⎛ π ⎞ a. ⎜4, ⎟ ⎝ 2 ⎠ a. (0, 4) b.(−3,π ) b.(3, 0) ⎛ 5π ⎞ c.⎜16, ⎟ ⎝ 6 ⎠ c.(−8 3,8) ⎛ π ⎞ d.⎜− 2,− ⎟ d. −1,1 ⎝ 4 ⎠ ( ) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 4 Example Rectangular and Polar Coordinates Find two different sets of polar coordinates for the point with the rectangular coordinate: A point has infinitely many sets a.(0, 4) of polar coordinates: b.(−3,3) a.(1,0),(1,2π ) c.(0,−4) ⎛ 3π ⎞ ⎛ π ⎞ b.⎜3 2, ⎟,⎜−3 2,− ⎟ d.(1, 3) ⎝ 4 ⎠ ⎝ 4 ⎠ ⎛ π ⎞ ⎛ 3π ⎞ c.⎜4,− ⎟,⎜4, ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛ π ⎞ ⎛ 4π ⎞ d.⎜2, ⎟,⎜−2, ⎟ ⎝ 3 ⎠ ⎝ 3 ⎠ Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 5 Circles Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 6 Rose Curves Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 7 Limaçon Curves Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 8 Lemniscate Curves Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 9 Spiral of Archimedes Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 10 Assignment: p. 564 #1-34 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 11 Parametric Equations of Polar Curves The polar graph of r = f (θ ) is the curve defined parametrically by: x = r cosθ = f (θ )cosθ y = r sinθ = f (θ )sinθ Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 12 Example Finding Slope of a Polar Curve Find the slope of the rose curve r = 2sin 3θ at the point where θ = π / 6. d 2sin 3θ sinθ dy dy / dθ ( ) = = dθ = − 3 π π d dx θ = dx / dθ θ = 6 6 (2sin 3θ cosθ ) π d θ = θ 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 13 Find the slope of the rose curve r = 2sin 3θ at the point where θ = π / 6. x = 2sin3θ cosθ y = 2sin3θ sinθ dy 6cos3θ sinθ + 2sin3θ cosθ = dx 6cos3θ cosθ − 2sin3θ sinθ 3cos3θ sinθ +sin3θ cosθ = 3cos3θ cosθ −sin3θ sinθ ⎛ 1 ⎞ ⎛ 3 ⎞ 3(0)⎜ ⎟+1⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ = ⎛ 3 ⎞ ⎛ 1 ⎞ = − 3 0⎜ ⎟−1⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 14 Area in Polar Coordinates The area of the region between the origin and the curve r = f (θ ) for α ≤ θ ≤ β is 2 β 1β 1 β 1 β 1 2 AA r 2 rd2d f ( f)( d) .d . == ∫α∫ θ θ= =∫α ∫( θ( )θ θ) θ α2 2 2α 2 € Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 15 Example Finding Area F ind the area of the region in the plane enclosed by the cardioid r = 2(1+ cosθ ). 2π 2π 1 1 2 A = ∫ r2 dθ = ∫ ⋅ 4(1+ cosθ) dθ 0 2 0 2 2π = ∫ 2(1+ 2cosθ + cos2 θ)dθ 0 2π = ∫ (3+ 4cosθ + cos2θ)dθ 0 2π ⎡ sin2θ ⎤ = ⎢3θ + 4sinθ + ⎥ = 6π − 0 = 6π ⎣ 2 ⎦0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 16 Area Between Polar Curves The area of the region between r1 (θ ) and r2 (θ ) for α ≤ θ ≤ β is β 2 β 2 β β 1 1 2 β 1 1 2 β 1 1 AA = r (r ()θ)d dθ − r ( r)(θd) dθ = r 2 rr22 −dr2. dθ. = ∫α∫ ( 2( θ2 ) ) θ − ∫α ∫ ( 1 θ( 1 ) )θ = ∫α ∫( 2 − 21 ) θ1 α2 2 2α 2 2 α 2 ( ) € Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 17 Area Between Polar Curves Find the area of the region lying between the inner and outer loops of the limacon r = 1− 2sinθ The inner loop begins and ends when r = 0. € 1 1− 2sinθ = 0 ⇒ sinθ = 2 π 5π ⇒ θ = , 6 6 € Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 18 Area Between Polar Curves Find the area of the region lying between the inner and outer loops of the limacon r = 1− 2sinθ The inner loop is traced as θ increases from π/6 to 5π/6. 1 5π/6 2 1 5π/6 A = (1− 2sinθ) dθ = 1− 4sinθ + 4sin2 θ dθ 2 ∫ π/6 2 ∫ π/6 ( ) 1 5π/6 ⎛ ⎛1− cos2θ ⎞⎞ 1 5π/6 € = 1− 4sinθ + 4 dθ = 1− 4sinθ + 2 1− cos2θ dθ ∫ π/6 ⎜ ⎜ ⎟⎟ ∫ π/6 ( ( )) 2 ⎝ ⎝ €2 ⎠⎠ 2 1 5π/6 = (3− 4sinθ − 2cos2θ)dθ 2 ∫ π/6 1 5π/6 = [3θ + 4cosθ −sin2θ] 2 π/6 ⎡⎛ ⎛ ⎞ ⎛ ⎞⎞ ⎛ ⎛ ⎞ ⎞⎤ 1 5π − 3 3 π 3 3 1 ⎡ ⎤ = ⎢⎜ + 4⎜ ⎟−⎜− ⎟⎟−⎜ + 4⎜ ⎟− ⎟⎥ = ⎣2π − 3 3⎦ 2⎣⎢⎝ 2 ⎝ 2 ⎠ ⎝ 2 ⎠⎠ ⎝ 2 ⎝ 2 ⎠ 2 ⎠⎦⎥ 2 3 3 = π − 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 19 Area Between Polar Curves Find the area of the region lying between the inner and outer loops of the limacon r = 1− 2sinθ The outer loop is traced as θ increases from 5π/6 to 13π/6. 1 13π/6 2 1 13π/6 A = (1− 2sinθ) dθ = 1− 4sinθ + 4sin2 θ dθ 2 ∫ 5π/6 2 ∫ 5π/6 ( ) 1 13π/6 ⎛ ⎛1− cos2θ ⎞⎞ 1 13π/6 € = 1− 4sinθ + 4 dθ = 1− 4sinθ + 2 1− cos2θ dθ ∫ 5 /6 ⎜ ⎜ ⎟⎟ ∫ 5 /6 ( ( )) 2 π ⎝ ⎝ 2€ ⎠⎠ 2 π 1 13π/6 = (3− 4sinθ − 2cos2θ)dθ 2 ∫ 5π/6 1 13π/6 = [3θ + 4cosθ −sin2θ] 2 5π/6 ⎡⎛ ⎛ ⎞ ⎛ ⎞⎞ ⎛ ⎛ ⎞ ⎛ ⎞⎞⎤ 1 13π 3 3 5π 3 3 1 ⎡ ⎤ = ⎢⎜ + 4⎜ ⎟−⎜ ⎟⎟−⎜ + 4⎜− ⎟−⎜− ⎟⎟⎥ = ⎣4π + 3 3⎦ 2⎣⎢⎝ 2 ⎝ 2 ⎠ ⎝ 2 ⎠⎠ ⎝ 2 ⎝ 2 ⎠ ⎝ 2 ⎠⎠⎦⎥ 2 3 3 = 2π + 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 20 Area Between Polar Curves Find the area of the region lying between the inner and outer loops of the limacon r = 1− 2sinθ The area between the inner and outer loops will be € 3 3 ⎛ 3 3 ⎞ 2π + − ⎜ π − ⎟ = π + 3 3 ≈ 8.34 2 ⎜ 2 ⎟ ⎝ ⎠ € Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 21 Assignment: p. 564 #39-53 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 22 Quick Quiz Sections 10.1 – 10.3 Y ou may use a graphing calculator to solve the following problems. 1. Which of the following is equal to the area of the region inside the polar curve r = 2cosθ and outside the polar curve r = cosθ? / 2 π / 2π 2 2 ((A)A) 3 30 cocoss θθdθdθ ∫ ∫0 π 2 π 2 ((B)B) 3 3∫0 cocoss θθddθθ ∫0 33 π / 2π / 2 2 2 ((C)C) ∫0 cocoss θθddθθ € ∫0 22 π / 2π / 2 € ((D)D) 3 3∫0 cocossθθdθdθ ∫0 (E) 3 π cπos d (E) 3∫0 cosθθ dθθ € ∫0 € € Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 23 Quick Quiz Sections 10.1 – 10.3 Y ou may use a graphing calculator to solve the following problems. 1. Which of the following is equal to the area of the region inside the polar curve r = 2cosθ and outside the polar curve r = cosθ? π / 2π / 2 2 2 ((A)A) 3 30 cocoss θθdθdθ ∫ ∫0 π 2 (B) 3 cπos 2 d (B) 3∫0 cosθθ dθθ ∫0 3 π / 2 3 π / 2 2 2 ((C)C) ∫0 cocoss θθddθθ € ∫0 22 π / 2π / 2 € ((D)D) 3 3∫0 cocossθθdθdθ ∫0 (E) 3 π cπos d (E) 3∫0 cosθθ dθθ € ∫0 € € Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 24 Quick Quiz Sections 10.1 – 10.3 2. For what values of t does the curve given by the parametric 3 2 44 22 eequations quations x = t − t −−11 aandnd yy = t ++22tt −−88tt hhaveave aa vverticalertical ttangent?angent? (A) 0 only (B) 1 only € (C) 0 and 2/3 only (D) 0, 2/3, and 1 (E) no value Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 25 Quick Quiz Sections 10.1 – 10.3 2. For what values of t does the curve given by the parametric 3 2 4 2 eequations quations x = t 3 − t 2 −−11 aandnd yy = t 4 ++22tt2 −−88tt hhaveave aa vverticalertical ttangent?angent? (A) 0 only (B) 1 only € (C) 0 and 2/3 only (D) 0, 2/3, and 1 (E) no value Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 10- 26 Quick Quiz Sections 10.1 – 10.3 3. The length of the path described by the parametric equations xx == t 2 and y = t from t = 0 to t = 4 is given by which integral? 4 4 ((A)A) 0 4t4+t +1d1tdt ∫ ∫0 4 2 (B) 2 4 t 2 1dt (B) 2∫0 t++1dt € ∫0 4 2 (C) ∫0 4 2t +1dt € (C) 2t 2 +1dt 4∫0 2 (D) ∫0 4t +1dt 4 2 € (D) 4 4t +21dt ∫0 (E) 2π ∫0 4t +1dt 4 € (E) 2 4t 2 +1dt ∫0 € € Copyright © 2007 Pearson Education, Inc.