1. What if we wanted the area enclosed by:
Or the slope of this tangent line?

2. Let’s start with the slope which for any curve is equal to
But how does one find this in relation to polar functions?
Here’s what we know:
How?
Product Rule, of course

3. To find the slope of a polar curve:

4. Find the slope of this tangent line at

5. Find the equation of this tangent line at

6. Find the equation of this tangent line at

7. What about the area enclosed by:
Start with this slice

8. What about the area enclosed by:
This slice is approximately the shape of an isosceles triangle
The area requires that we know height and base.
Remember arclength from trig?
where  must be in radians
In this case, the angle is very small so we will call it d
So the area of this slice can be approximated as

9. What about the area enclosed by:
To extend this approximation over the whole region we will need an…
Integral
I knew you wouldn’t forget
 and  are usually 0 and 2p but it depends upon the problem

10. Now find the area…
Example: Find the area enclosed by:

12. The shaded region to the right is bounded by the polar curves:
But what about this one?
The shaded region to the right is bounded by the polar curves:
and
Find the area of the shaded region.
We need to find  and .
So we need to set the two curves equal to each other.

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

14. Using Symmetry
Area of one leaf times 4:
Area of four leaves:

15. p 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: p