Day 49 in MAT229

Today:

• Announcements:
  • You have an assignment due soon (HW 9).

  • One more comment about the exam: Key

  • HW 8 is returned: a few comments
    1. Don't forget to check the endpoints of the interval!
    2. Check your initial value for your summation for the derivative: does it give the proper first term?
    3. Multiplying power series.

  • On Friday of this week there's a set of four (short) talks, by undergraduates: Come if you can!

• Today:
  • Let's wrap up Sect. 10.2: Calculus of parametric equations (pdf, pdf summary)

    • Curve (Arc) Length

      Suppose a particle follows 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, $d=r t$ (in its modified form $\Delta d = r \Delta t$).

      In this case, the rate is just the speed. You might not be too surprised by the speed formula in two dimensions: in one dimension, it is \[ r = \left| x'(t_i) \right| = \sqrt{x'(t_i)^2} \] In two dimensions, \[ r = \sqrt{x'(t_i)^2+y'(t_i)^2}. \] 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,

      \[ D=\int_a^b\sqrt{x'(t)^2+y'(t)^2} dt \]

  • Then it's on to polar coordinates: Sect. 10.3: Polar coordinates (pdf, pdf summary)