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

Length in Polar Coordinates

Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.

r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2

Identifying Conic Sections

Complete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

x² + y² + 4x + 2y = 1

Identifying Parametric Equations in the Plane

Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

x = √t, y = 1 − √t, t ≥ 0

Lengths of Curves

Find the lengths of the curves in Exercises 13–19.

x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2

Cartesian to Polar Equations

Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.

x² + y² + 5y = 0

Polar to Cartesian Equations

Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.

r cos (θ − 3π/4) = (√2)/2