Area in Polar Coordinates Lesson 10.10

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

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

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

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

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

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

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

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

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