1. 10.6: The Calculus of Polar Curves
Try graphing this on the TI-89.
Greg Kelly, Hanford High School, Richland, Washington

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

3. To find the slope of a polar curve:

4. Example:

5. Area Inside a Polar Graph:
The length of an arc (in a circle) is given by r. q when q is given in radians.
For a very small q, the curve could be approximated by a straight line and the area could be found using the triangle formula:

6. We can use this to find the area inside a polar graph.

7. Example: Find the area enclosed by:

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

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

11. 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:

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