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 dθdθ α β

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

5. Guidelines 1.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 2.Sketch a typical circular sector Label central angle dθ 3.Express the area of the sector as 4.Integrate the expression over the limits from a to b

6. Find the Area Given r = 4 + sin θ  Find the area of the region enclosed by the ellipse dθdθ 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 dθdθ

10. Assignment Lesson 10.10A Page 459 Exercises 1 – 19 odd Lesson 10.10B Page 459 Exercises 21 – 27 odd