18.01 Single Variable Calculus Fall 2006

Transcription

1 MIT OpenCouseWae Single Vaiable Calculus Fall 006 Fo infomation about citing these mateials o ou Tems of Use, visit:

2 Lectue 3: Pola Co-odinates, Aea in Pola Co-odinates Pola Coodinates θ Figue : Pola Co-odinates. In pola coodinates, we specify an object s position in tems of its distance fom the oigin and the angle θ that the ay fom the oigin to the point makes with espect to the x-axis. Example. What ae the pola coodinates fo the point specified by (, ) in ectangula coodinates? (,-) Figue : Rectangula Co-odinates to Pola Co-odinates. = + ( ) = θ = 4 In most cases, we use the convention that 0 and 0 θ. But anothe common convention is to say 0 and θ. All values of θ and even negative values of can be used.

3 θ x y Figue 3: Rectangula Co-odinates to Pola Co-odinates. Regadless of whethe we allow positive o negative values of o θ, what is always tue is: x = cos θ and y = sin θ Fo instance, x =, y = can be epesented by =, θ = 3 : 4 = x = cos 3 and = y = sin Example. Conside a cicle of adius a with its cente at x = a, y = 0. We want to find an equation that elates to θ. (a,0) Figue 4: Cicle of adius a with cente at x = a, y = 0.

4 We know the equation fo the cicle in ectangula coodinates is (x a) + y = a Stat by plugging in: x = cos θ and y = sin θ This gives us ( cos θ a) + ( sin θ) = a cos θ acosθ + a + sin θ = a a cos θ = 0 = a cos θ The ange of 0 θ taces out the top half of the cicle, while θ 0 taces out the bottom half. Let s gaph this. y θ = /4 θ (a,0) θ = 0 x Figue 5: = a cos θ, / θ /. At θ = 0, = a = x = a, y = 0 At θ =, = a cos = a 4 4 The main issue is finding the ange of θ tacing the cicle once. In this case, < θ <. θ = (down) θ = (up) 3 Weid ange (avoid this one): < θ <. When θ =, = a cos = a( ) = a. The 3 adius points backwads. In the ange < θ <, the same cicle is taced out a second time. 3

5 =f(θ) Figue 6: Using pola co-odinates to find aea of a geneic function. Aea in Pola Coodinates Since adius is a function of angle ( = f(θ)), we will integate with espect to θ. The question is: what, exactly, should we integate? θ?? dθ Let s look at a vey small slice of this egion: θ dθ dθ Figue 7: Appoximate slice of aea in pola coodinates. This infinitesimal slice is appoximately a ight tiangle. To find its aea, we take: Aea of slice (base) (height) = ( dθ) So, θ Total Aea = dθ θ 4

6 Example 3. = a cos θ, and < θ < (the cicle in Figue 5). / / A = aea = (a cos θ) dθ = a cos θ dθ / / Because cos θ = + cos θ, we can ewite this as / / / A = aea = ( + cos θ) dθ = a dθ + a cos θ dθ / / / / 0 = a + sin θ = a + [sin sin( )] / A = aea = a Example 4: Cicle centeed at the Oigin. =a Figue 8: Example 4: Cicle centeed at the oigin x = cos θ; y = sin θ x + y = cos θ + sin θ = The cicle is x + y = a, so = a and x = a cos θ; y = a sin θ A = a dθ = a = a. 0 5

7 Example 5: A Ray. In this case, θ = b. θ=b Figue 9: Example 5: The ay θ = b, 0 <. The ange of is 0 < ; x = cos b; y = sin b. Example 6: Finding the Pola Fomula, based on the Catesian Fomula y /sin θ θ x Figue 0: Example 6: Catesian Fom to Pola Fom Conside, in catesian coodinates, the line y =. To find the pola coodinate equation, plug in y = sin θ and x = cos θ and solve fo. sin θ = = = with 0 < θ < sin θ 6

8 Example 7: Going back to (x, y) coodinates fom = f(θ). Stat with =. + sin θ Hence, Plug in = x + y : Useful convesion fomulas: ( ) = x + y and θ = tan y x + sin θ = x + y + y = ( ) x + y = y = x + y = y Finally, 3y x + + y = 4 This is an equation fo an ellipse, with the oigin at one focus. = y + y 4 Example 8: A Rose = cos(θ) The gaph looks a bit like a flowe: <0 /4 >0 >0 <0 -/4 Figue : Example 8: Rose Fo the fist petal < θ < 4 4 Note: Next lectue is Lectue 34 as Lectue 33 is Exam 4. 7