16. Parametric Equations & Polar Coordinates

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

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

Find the slope of the tangent line of the polar curve $r=2\cos \theta$ at $\theta =\frac{\pi }{3}$.

Finding Polar Areas

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

Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

Area in Polar Coordinates

Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.

Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ

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

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

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

The region inside the circle r = 8 sin θ

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

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.

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

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

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 2θ; (π/2, π/4)

25–28. Horizontal and vertical tangents Find the points at which the following polar curves have horizontal or vertical tangent lines.

r = 4 cos θ

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 1 - sin θ; (1/2, π/6)

Find the area enclosed by the cardioid $r=2-2\sin \theta$ between $\theta =\frac{\pi }{6}$ and $\theta =\frac{2\pi }{3}$.