16. Parametric Equations & Polar Coordinates

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 =3 − 6 cos θ

Finding Polar Areas

Find the areas of the regions in Exercises 9–18.

Shared by the circles r = 1 and r = 2 sin θ

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)

A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?

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

The limaçon r=3−6cosθ

Spiral arc length Consider the spiral r=4θ, for θ≥0.

c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.

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).

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves

r = 1 + sin θ and r = 1 + cos θ

45–60. Areas of regions Find the area of the following regions.

The region inside one leaf of the rose r = cos 5θ

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).

a. Find the area Aₙ of Rₙ.

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.

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside one leaf of r = cos 3θ

45–60. Areas of regions Find the area of the following regions.

The region outside the circle r = 1/2 and inside the circle r = cos θ

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.

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves

r = 3 sin θ and r = 3 cos θ

63–74. Arc length of polar curves Find the length of the following polar curves.

The spiral r = θ², for 0 ≤ θ ≤ 2π