1. Area in Polar Coordinates
Lesson 10.10

2. Area of a Sector of a Circle
Given a circle with radius = r
Sector of the circle with angle = θ
The area of the sector given by θ r

3. Area of a Sector of a Region
Consider a region bounded by r = f(θ)
A small portion (a sector with angle dθ) has area β • dθ • α

4. Area of a Sector of a Region
We use an integral to sum the small pie slices β
r = f(θ) • • α

5. Guidelines Use the calculator to graph the region
Find smallest value θ = a, and largest value θ = b for the points (r, θ) in the region
Sketch a typical circular sector
Label central angle dθ
Express the area of the sector as
Integrate the expression over the limits from a to b

6. The ellipse is traced out by 0 < θ < 2π
Find the Area
Given r = 4 + sin θ
Find the area of the region enclosed by the ellipse dθ
The ellipse is traced out by 0 < θ < 2π

7. Areas of Portions of a Region
Given r = 4 sin θ and rays θ = 0, θ = π/3
The angle of the rays specifies the limits of the integration

8. Area of a Single Loop Consider r = sin 6θ Note 12 petals
θ goes from 0 to 2π
One loop goes from 0 to π/6

9. Area Of Intersection
Note the area that is inside r = 2 sin θ and outside r = 1
Find intersections
Consider sector for a dθ
Must subtract two sectors dθ

10. Assignment Lesson 10.10A Page 459 Exercises 1 – 19 odd Lesson 10.10B