1. 10.6B and 10.7
Calculus of Polar Curves

2. Try graphing this on the TI-Nspire

3. Graphing Polar Equations Recognizing Common Forms
Circles
Centered at the origin: r = a
radius: a period = 360
Tangent to the x-axis at the origin: r = a sin 
center: (a/2, 90) radius: a/2 period = 180
a > 0  above a < 0  below
Tangent to the y-axis at the origin: r = a cos 
a > 0  right a < 0  left
r = 4 sin
r = 4 cos
Note the Symmetries

4. Graphing Polar Equations Recognizing Common Forms
Flowers (centered at the origin)
r = a cos n or r = a sin n
radius: |a|
n is even  2n petals
petal every 180/n
period = 360
n is odd  n petals
petal every 360/n
period = 180
cos  1st 0
sin  1st 90/n
r = 4 sin 2
r = 4 cos 3
Note the Symmetries

5. Graphing Polar Equations Recognizing Common Forms
Spirals
Spiral of Archimedes: r = k
|k| large  loose |k| small  tight
r = 
r = ¼ 

6. Graphing Polar Equations Recognizing Common Forms
Heart (actually: cardioid if a = b … otherwise: limaçon)
r = a ± b cos  or r = a ± b sin 
r = cos 
r = cos 
r = sin 
r = sin 
Note the Symmetries

7. Graphing Polar Equations Recognizing Common Forms
Leminscate
a = 16
Note the Symmetries

8. To find the slope of a polar curve:
We use the product rule here.

9. To find the slope of a polar curve:

10. Example:

11. Area Inside a Polar Graph:

12. Tangent lines at the pole
The line is tangent to the graph of
at the pole if
Ex. Graph and find the tangent(s) at the pole

13. Example: Find the area enclosed by:

15. Notes:
To find the area between curves, subtract:
Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

18. When finding area, negative values of r cancel out:
Area of one leaf times 4:
Area of four leaves:

19. To find the length of a curve:
Remember:
For polar graphs:
If we find derivatives and plug them into the formula, we (eventually) get:
So:

20. There is also a surface area equation similar to the others we are already familiar with:
When rotated about the x-axis: p