section 3.2 – polar equations section 3.3 – polar graphs
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Section 3.2 – Polar EquationsSection 3.3 – Polar Graphs
Shapes of Polar Curves
Graphing Polar Curves on Calculator
Finding Points of Intersection(Boundaries of Integration)
The Shapes
Lines
Circles
Cardioid
Lemniscate
Sprial
Rose Curves
The Shapes - Linesr cos 3
Vertical Line
r sin 3 Horizontal Line
r 2cos 5sin 3 0 General Line
The Shapes - Circles
r 5
r 2cos
r 4sin
The Shapes – Cardioids(Hearts)
r
r 2 4sin
4 2cos
r
r a bsin
a bcos
Note: Extra loopOnly if b > a
The Shapes – Lemniscates(Propellers)
2 2r a cos2 2 2r a sin2
r 4cos2 r 5sin2
13
The Shapes - Spirals
r 2 0,2
r a
r 0.3 0,12
The Shapes – Rose Curves
r 5cos3 r 6sin4
r asin n
r bsin n
n even – 2n pedalsn odd – n pedals
Graphing Polar Curves on Calculator
1. Change mode to Polar
2. Hit y = (you’ll see r = indicating polar mode)
3. Enter the equation 3
r 5cos2
4. Graphing once should give you a sense of how to change the x, y and theta constraints.
5. Now change the constraints in WINDOW
min 2
max 2
Xmin 5
Xmax 5
Xscl 1
Ymin 5
Ymax 5
Yscl 1
Finding Points of Intersection(Boundaries of Integration)
2 2r sin and r cos
sin cos
tan 1
5,
4 4
2 2r
2
4 42 8r
22
4 4 4 48 8 8 5 8 5, , , , , , ,
2 4 2 4 2 4 2 4
r 1and r 2sin2
1sin2
2
5 13 172 , , ,
6 6 6 6
5 13 17, , ,
12 12 12 12
5 13 171, , 1, , 1, , 1,
12 12 12 12
5 7 11 13 17 19 231, , 1, , 1, , 1, , 1, , 1, , 1, , 1,
12 12 12 12 12 12 12 12
WRONG ANSWER
r 1 cos and r 1 sin2 2
tan 12
3 7 3,
2 4 4 2
r 1 sin2
3r 1 sin
4
2r 1
2
2 2 31 , , 1 ,
2 2 2 2
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