16. Parametric Equations & Polar Coordinates

Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).

c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).

e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).

84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is

L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.

40–41. {Use of Tech} Slopes of tangent lines

b. Find the slope of the lines tangent to the curve at the origin (when relevant).

r = 1 −sin θ

51–52. {Use of Tech} Arc length of polar curves Find the approximate length of the following curves.

The limaçon r=3−6cosθ

44–49. Areas of regions Find the area of the following regions.

The region inside the limaçon r=2+cosθ and outside the circle r=2