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

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.

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

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

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

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

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)

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

r = 4 + sin θ; (4, 0) and (3, 3π/2)