Finding Polar Areas


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


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

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

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

The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ

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

Finding Lengths of Polar Curves

Find the lengths of the curves in Exercises 21–28.

The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4

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

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