1. Section 12.6 – Area and Arclength in Polar Coordinates
12.2

2. Integrals in Polar Coordinates
Area of a Polar Curve:
These are needed to make some integration possible.

3. The total area of the region enclosed
NO CALCULATOR
by the polar graph of is

4. NO CALCULATOR Not an option. Let’s check the graph.
Symmetric about the x-axis. So...

5. CALCULATOR REQUIRED
The area of the region enclosed by the graph of the polar
curve is
4.712
9.424
18.849
37.699
75.398

6. CALCULATOR REQUIRED
The approximate total area of the region enclosed by the
polar graph of is:
0.393
0.785
1.178
1.571
1.873

7. Find the area inside
It’s a circle!
Using calculus:
Wait! What?!?
The circle actually gets drawn twice. We need to change our limits of integration.
Better.

8. CALCULATOR REQUIRED
Set up to definite integral to find the area inside the smaller
loop of

9. Or we could restrict it to exclude the loop.
If would have wanted the total area, we would have to change things a bit.
Total Area.
Inner Loop.
Without the loop.
We could take the whole area and subtract the inner loop (no double counting).
Or we could restrict it to exclude the loop.

10. CALCULATOR REQUIRED
Find the area inside
What?!?
Let’s try another way.

11. NO CALCULATOR: Find the area of the region inside the
circle r = 4 and outside

12. We’ll need to subtract red from blue.
=7.653

13. =0.571 We’ll need to add red and blue because they make up two halves.
However, the angle ranges are different for the two functions.
=0.571

14. Length of an Arc in Polar Coordinates

15. CALCULATOR REQUIRED

16. NO CALCULATOR