Polar Coordinates
 Give 4 different pairs of polar coordinates thatcorrespond to the Cartesian Coordinates (2,2)
Aim: How do we describe Curves in Polar form?
http://www.glossyart.net/tutorial/images/polar_coordinates.gif
Definition
A Polar Coordinate System isa method of locating a point(r,   ) in a plane where r is adistance from a point calledthe pole directed at an angleθ  measuredcounterclockwise from apolar axis.
Example
[image]
Polar Grid
Pole
Polar Axis
r=10
(r,θ )=
[image]
r=10
r= ̶ 10
Negative radius means go in theopposite directions
Rememberto considerthe rotationFIRST
Example
[image]
(r,θ )=
Find thepolarcoordinates:
Example
[image]
(r,θ )=
Find thepolarcoordinates:
[image]
Plot
Name the point with three other polarcoordinates using
Important Idea
[image]
Polar coordinates         arerelated to rectangularcoordinates         as follows:
θ
r
y
x
Example
Graph:
[image]
r=6
What about
r = ̶  6?
Example
[image]
Graph:
θ=
What about
Some ClassicPolar Graphs
Example
limacon
[image]
A ClassicPolar Graph
Example
Spiral ofArchimedes
[image]
Some ClassicPolar Graphs
[image]
Example
8 petal rose
Use yourgraphingcalculatorto graph
MODE
POL
GRAPH
Y=
CARDIOID
[image]
Polar Roses
Important Idea
8
4 petals
[image]
Polar Roses
Important Idea
10
8 petals
[image]
Polar Roses
Try This
Name theangle  in rads?
[image]
[image]
Important Idea
What do you observe about thenumber of petals? The period?
[image]
Some Calculus
Find the slope of
at            .
We can write polar equations parametrically
where
What is this telling us aboutthe curve?
Area
r
S
 
in radians
[image]
Analysis
Find the areafrom 0 to
0
Area1=
[image]
Analysis
Find the areafrom 0 to
0
Area2=
[image]
Analysis
Find the areafrom 0 to
0
Area3=
[image]
Analysis
Find the areafrom 0 to
0
Riemann Sum for total area
[image]
Analysis
Find the areafrom 0 to
0