W14 Regular

Due date: Sunday 4/19, 11:59pm

Polar curves

01

05

Sketching limaçons

Sketch the graphs of the following polar functions:

(a) $r=2+\sin \theta$ $$ (b) $r=2\cos \theta$

(c) $r=1+2\sin \theta$ $$ (d) $r=3+3\cos \theta$

Link to original

Solution

Solutions---0270-05

Polar calculus

02 $★$

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.

Link to original

Solution

Solutions---0280-03

03

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.

Link to original

Solution

Solutions---0280-02

04 $★$

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.

Link to original

Solution

Solutions---0280-06

05 $★$

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.

Link to original

Solution

Solutions---0280-05