1. 10.3 day 2 Calculus of Polar Curves Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Lady Bird Johnson Grove, Redwood National Park, California

2. Try graphing this on the TI-84.

3. Polar-Rectangular Conversion Formulas

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

5. To find the slope of a polar curve:

6. Example:

7. Find the slope of the curve at the given values. Find the points where the curve has horizonal or vertical tangent lines.

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

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

10. Example: Find the area enclosed by:

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

13. Find the area of the region that lies inside the circle r = 3 and outside the cardioid.

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

15. Find the area that lies outside the four-petal rose and inside the circle.

16. 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:

17. Or… Convert to Parametric! Find the length of the cardioid

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