Problems

01

Polar curve - Vertical or horizontal tangent lines

Find all points on the given curve where the tangent line is horizontal or vertical.

$$r=\cos \theta \theta \in [\mathrm{0,2}\pi )$$

Hint: First determine parametric Cartesian coordinate functions using $\theta$ as the parameter.

02

Arclength of one loop of a rose

Consider the graph of the polar curve $r=2\cos 3\theta$.

Set up an integral which computes the arclength of one loop of this curve.

03

Polar curve - Slope of tangent line

Find the slope of the tangent line to the given polar curve:

$$r=\sin \theta +3\cos \theta \text{at }\theta =\pi /2$$

Hint: First determine parametric Cartesian coordinate functions using $\theta$ as the parameter.

04

Polar coordinates - lunar areas

(a) Find the area of the green region.

(b) Find the area of the yellow region.

(You can find these in either order.)

05

Pickup region of a microphone - limaçon area

The pickup region of a microphone is described by a limaçon with equation $r=8+8\sin \theta$, and part of the region is on a stage.

Find the area of the part of the region on the stage.

06

Area of an inner loop

A limaçon is given as the graph of the polar curve $r=1+2\sin \theta$.

Find the area of the inner loop of this limaçon.