Day 28 in MAT227

Last time: More Section 12.1

Next time: Section 12.3

Today:

Announcements:
- Assignment 9.1 returned.
  - #4: how many used the hint? The other trick is implicit differentiation... it makes it easier.
  - #9: ugly stuff, but you can do it! It's algebra....
  - #26: we did in class
  - #44: just use your calculator
- Assignment 10.1 due (remember, revised due dates....)
- Your test corrections are due today: here's Exam 2 (if you want to make a copy for corrections)
Section 12.2: Arc Length and Speed
- Today's lab features a critical application of parametric curves: Planetary Orbits (ellipses and more!).
  This is our last lab: you will turn in your lab notebooks (include only relevant pages, and you may tear those out and put them in a manilla folder, for example) with the final exam.
  To get in the mood, let's have a look at Theorem 1, p. 639, from section 12.4: it informs us about ellipses.
  Then we'll have a look at the parameterization of an ellipse from section 12.1 (Example 4).
- The speed formula (our author should be short for using s for length, when defining speed -- which should be denoted by s!):
  
  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:
  - #1, p. 620
  - #13, p. 620; see page 617 for the equations of the cycloid.
  - #14, p. 620
  - #21, p. 620
  - #36, p. 621

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