Subsections

• Purpose
• Getting Started
• Background
  • Cardioids, Limaçons, and Roses
  • Intersections of Curves in Polar Coordinates
  • Area in Polar Coordinates
  
  
• Exercises

Graphs in Polar Coordinates

Purpose

The purpose of this lab is to help you become familiar with graphs in polar coordinates.

Getting Started

To assist you, there is a worksheet associated with this lab that you can copy into your home directory by going to your computer's start menu and choose run. In the run field type:

\\filer\calclab

when you hit enter, you can then choose MA1023 and then choose the worksheet

Polar_start_B11.mw

Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting the exercises.

Background

There are many times in math, science, and engineering that coordinate systems other than the familiar one of Cartesian coordinates are convenient. In this lab, we consider one of the most common and useful such systems, that of polar coordinates.

The main reason for using polar coordinates is that they can be used to simply describe regions in the plane that would be very difficult to describe using Cartesian coordinates. For example, graphing the circle $x^2+y^2=1$ in Cartesian coordinates requires two functions - one for the upper half and one for the lower half. In polar coordinates, the same circle has the very simple representation $r=1$.

Cardioids, Limaçons, and Roses

These are three types of well-known graphs in polar coordinates. The table below will allow you to identify the graphs in the exercises.

Name	Equation
cardioid	$r = a(1 \pm \cos(\theta))$ or $r = a(1 \pm \sin(\theta))$
limaçon	$r = b \pm a \cos(\theta)$ or $r = b \pm a \sin(\theta)$
rose	$r=a \cos(n \theta)$ or $r=a \sin(n \theta)$

Intersections of Curves in Polar Coordinates

Finding where two graphs in Cartesian coordinates intersect is straightforward. You just set the two functions equal and solve for the values of $x$. In polar coordinates, the situation is more difficult. Most of the difficulties are due to the following considerations.

1. A point in the plane can have more than one representation in polar coordinates. For example, $r=1$, $\theta = \pi$ is the same point as $r=-1$, $\theta=0$. In general a point in the plane can have an infinite number of representations in polar coordinates, just by adding multiples of $2\pi$ to $\theta$. Even if you restrict $\theta$ a point in the plane can have several different representations.
2. The origin is determined by $r=0$. The angle $\theta$ can have any value.

These considerations can make finding the intersections of two graphs in polar coordinates a difficult task. As the exercises demonstrate, it usually requires a combination of plots and solving equations to find all of the intersections.

Area in Polar Coordinates

The relationship between area and integrals in polar coordinates is a little strange; the area inside a circle given (in polar coordinates) by $r=a$ is NOT just $$\int_{0}^{2 \pi}r \, d \theta$$. Here is the rule: Area inside $r=f(\theta)$ is given by $$\frac{1}{2}\int_{\theta_1}^{\theta_2}r^2 \, d \theta$$= $$\frac{1}{2}\int_{\theta_1}^{\theta_2} f(\theta)^2 \, d \theta$$. Note that this gives you the right answer for a circle: $A=\pi r^2$. So to find the area of the cardiod use the following command.

>Area1:=1/2*int((1-cos(theta))^2, theta=0..2*Pi);
>evalf(Area1);

Exercises

1. For each of the following polar equations, plot the graph in polar coordinates using the plot command and identify the graph as a cardioid, limaçon, or rose.
   1. $r = \cos(2 \theta)$
   2. $r = 1 +2 \sin(\theta)$
   3. $r = 1 + \cos(\theta+\pi/4)$
   4. $r = 4\sin(4\theta)$

2. Find all points of intersection for each pair of curves in polar coordinates.
   1. $r = 1+\cos(\theta)$ and $r=3/2$ for $0 \leq \theta \leq 2\pi$.
   2. $r = 1-\sin(\theta)$ and $r=3/2+\cos(\theta)$ for $0 \leq \theta \leq 2\pi$. Find the area inside $r = 1-\sin(\theta)$ and outside $r=3/2+\cos(\theta)$.

3. Plot the polar equation $r=1+3\cos(\theta)$. Find the angles that create the the inner loop of $r=1+3\cos(\theta)$. Plot only the inner loop and then find the area inside the inner loop.