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