Day 40, Mat227

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Announcements:
- Your final homework is due today.
- Today we wrap up polar coordinates, with some calculus of polar curves.
  You won't have a homework on these, but might be expected to reproduce some of those we'll examine together (or examples from the sections) for the final exam.
Last time we covered Section 10.3: Polar Coordinates
We can think of every point on the plane as being given as the intersection of a vertical and a horizontal line. These lines give rise to the xy-coordinates of a point.
Alternatively, you can think of every point in the plane being given by the intersection of a circle centered at the origin, with given radius ${r}$ , and of a ray coming out from the origin at a given angle ${ \theta}$ from the positive x-axis (the "polar axis"). These coordinates give rise to the polar representation of the plane.
If you look over the pictures in this section, you will see that there are many beautiful curves that one can create using polar coordinates and polar plots.
See, for example, p. 685 -- which should remind you of a spirograph, if any of you are that old....
The polar coordinate system was introduced by Newton, an alternative to the Cartesian coordinate system. It is especially useful for constructing or representing objects which exhibit radial symmetry (e.g. a circle).
In the polar coordinate system every point in the plane is expressed by its distance and direction (angle) from the origin (called the pole). The polar axis plays the role formerly played by the positive x-axis. The polar coordinates are often given as ${(r,\theta)}$ , where
- r is the distance from the pole (although we may allow r to be negative), and
- ${\theta}$ is the angle between the polar axis and the ray passing through the point of interest.
Here it is:

We represent the point P in one of two ways -- in rectangular coordinates (x,y), or in polar coordinates ${(r,\theta)}$ . We can represent any point of the plane in either of those ways. Note, however, that we have a certain level of redundancy in the polar coordinate system.
Look at the origin, for example.
Furthermore, we think of ${\theta}$ as an angle, and the plane is obviously then represented in a ${2{\pi}}$ -periodic way.
Suppose we allow r to be negative in the image above: then we may think of the point ${P}$ above as the point ${(-r,\theta+\pi)}$ (for example), or ${(-r,\theta+3\pi)}$ for that matter.
So now curves are given in terms of r and ${\theta}$ rather than x and y, via
- ${x={r\cos({\theta})}}$
- ${y={r\sin({\theta})}}$
Alternatively, we can write r and in terms of x and y:
- ${r^2={{x^2}+{y^2}}}$
- ${\tan(\theta)={\frac{y}{x}}}$
(but notice that ${r}$ and ${ \theta}$ are both given implicitly).
Since we just finished parametric curves, we might think of polar coordinates as examples of parametric curves, in the following way: we will often be given , and then can think of x and y as functions of :
- ${x={f(\theta)\cos({\theta})}}$
- ${y={f(\theta)\sin({\theta})}}$
with as the parameter.
Examples:
- #1, p. 686
- #3
- #10
- #16
- #27
- #67
- #47
- #56
- #64
Section 10.4: Areas and Lengths in Polar Coordinates
- Length:
  The easiest way to get the length of a polar curve is to use parametrics. Let's do it that way.
  ${D=\int_a^b\sqrt{x'(t)^2+y'(t)^2}dt}$
  becomes
  
  ${D=\int_a^b\sqrt{r'(t)^2+r(t)^2}dt}$
  Let's see how....
- Area:
  For the area, let's start with a circle of radius r, centered at the origin, and see how we might adapt the "familiar formulas" to more exotic shapes, until we obtain
  
  ${A=\int_a^b\frac{1}{2}r(\theta)^2d\theta}$
  The derivation relies on understanding when one thing ( ${A}$ ) is a linear function of another thing ( ${\theta }$ -- in the case of a circle).
Now, to some exercises:
- #5, p. 692
- 7
- 15
- 26
- 30
- 47
- 54

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