Day 46 in MAT229

Last time: Section 12.2

Next time: Section 12.3/12.4

Today:

Announcements:
- Section 12.1 returned: graded #12 and 40.
- Sean recommends "justmathtutoring.com" videos for help.
- Section 12.2 problems are due tomorrow.
- Change in plans: Section 12.3 problems: one of them will appear on the exam -- okay?
Section 12.2: Arc Length and Speed
- I already indicated above what the speed should be -- so I'm not going to spend much time on that.
  How can we use this formula, however? Suppose a particle has followed the parametric curve C(t)=(x(t),y(t)): then we can compute how far the particle has travelled during the interval ${a\le{t}\le{b}}$ easily using the dirt formula, d=rt (in its modified form ${\Delta{d}=r\Delta{t}}$ ).
  In this case, the rate is just the speed. So we compute the integral
  
  ${D\approx\sum_{i}^{}\Delta{d_i}=\sum_{i}^{}\sqrt{x'(t_i)^2+y'(t_i)^2}\Delta{t_i}}$
  
  or, in the limit,
  
  ${D=\int_{a}^{b}\sqrt{x'(t)^2+y'(t)^2}dt}$
  This is actually just a re-expression of the arc length formula:
  
  ${D=\int_{x_0}^{x_1}\sqrt{1+y'(x)^2}dx=\int_{x_0}^{x_1}\sqrt{1+\left(\frac{y'(t)}{x'(t)}\right)^2}dx}$ and then we factor out the x'(t), to get ${D=\int_{a}^{b}\sqrt{x'(t)^2+y'(t)^2}dt}$ where ${x(a)=x_0}$ and ${x(b)=x_1}$ .
  But arc length may be different from the distance the particle travelled: a particle can revisit many sections of the curve y(x) -- so once again we need to be careful to distinguish between the independent variable of interest (whether x or t).
- Other calculations can be generalized. For example, the formula for surface area:
- Examples:
  - #2, p. 620
  - #13, p. 620; and let's start by developing the equations for the cycloid.
  - #14, p. 620
  - #21, p. 620
  - #36, p. 621
Section 12.3: Polar Coordinates
- Homework: one of these will appear on the exam: pp. 627-630, #4, 6, 8, 9, 10, 13, 14, 22, 25, 31, 50, 51
- We're going to be using a different coordinate system to describe points in the plane -- no big deal, except that you're quite likely unfamiliar with the new system (called polar coordinates):
  
  Coordinate r is called the radial coordinate and $\theta$ is called the angular coordinate.
  Notice something interesting: what are the radial coordinates of (x,y) and (-x,-y)?
  (See the constraints on $\theta$ on page 627)
- If horizontal and vertical lines are very easily expressed in the usual Cartesian coordinates system, it's circles and lines centered at the origin that are easily expressed in polar coordinates. Here are the "grid lines" in polar coordinates:
- One of the problems of polar coordinates is their non-uniqueness:
  
  ${(r,\theta)=(r,\theta+2n\pi)}$
  We can sometimes use that to our advantage, however; for example, if we want to imagine a point travelling around a circle, then $\theta$ is a parameter, and by letting it run over multiples of $2\pi$ the point makes multiple trips around the circle....
- We sometimes permit r to be negative, too, for convenience. In terms of non-uniqueness we also have that
  
  ${(r,\theta)=(-r,\theta+\pi)}$
- One of the advantages of polar coordinates is that they allow us to create curves that can't be represented as functions of Cartesian coordinates y(x).
- Examples:
  - Understanding polar coordinates:
    - #1, p. 627
    - #2, p. 627
  - Converting between coordinate systems:
    - #3, p. 627
    - #5, p. 628
    - #20, p. 628
  - Converting equations:
    - #16, p. 628
    - #20, p. 628
    - #33, p. 629
  - Deriving equations in polar coordinates:
    - a line passing through (x,y)=(a,b) with slope m

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