16. Parametric Equations & Polar Coordinates

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

The region inside the curve r = √(cos θ) and inside the circle r = 1/√2 in the first quadrant

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

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

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

The region inside the rose r = 4 sin 2θ and inside the circle r = 2

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

The region common to the circles r = 2 sin θ and r = 1

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

The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ

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

The region inside the limaçon r = 4 - 2 cos θ

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

The complete circle r = a sin θ, where a > 0

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

The spiral r = θ², for 0 ≤ θ ≤ 2π

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

The complete cardioid r = 4 + 4 sin θ

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

The curve r = sin³(θ/3), for 0 ≤ θ ≤ π/2

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

{Use of Tech} The complete limaçon r=4−2cosθ

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

a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.

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.

Circles in general Show that the polar equation

r² - 2r r₀ cos(θ - θ₀) = R² - r₀²

describes a circle of radius R whose center has polar coordinates (r₀, θ₀)

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.