10.5 Area and Arc Length in Polar CoordinatesMiss BattagliaAP Calculus

Area in Polar CoordinatesIf f is continuous and nonnegative on the interval [α,β], 0 < β – α < 2π, then the area of the region bounded by the graph of r=f(θ) between the radial lines θ=α and θ=β is given by

Proof of AreaRemember the area of a sector is given by ½θr2

Radius of ith sector = f(θi)

Central angle of ith sector = (β – α)/n = Δθ

Taking the limit as n ∞ produces

Find the area enclosed by one loop of r=sin(4θ)

Arc Length of a Polar CurveLet f be a function whose derivative is continuous on an interval α < θ < β. The length of the graph of r=f(θ) from θ=α to θ=β is

(Proof is Exercise 89 Section 10.5)

Find the arc length from θ=0 to θ=2π for the cardioid r=f(θ)=2-2cosθ

1. Find the area of one petal of the rose curve given by r=3cos(3θ)

2. Find the arc length from θ=0 to θ=π for the cardioid r=f(θ)=sin2(θ/2)

Classwork

Homework

•Read 10.5 Page 715 #1, 4, 7, 9, 11, 13, 25