Line Integrals

So on Tuesday, we continued the lesson on line intergrals. A few students (me being one of them) were caught cutting class on Monday. Luckily we weren't punished. At least not yet.
We reviewed the four step process for completing a line integral:
Start with the integral of f (x,y) ds
1) Find a parametric curve, including the limits of the curve
2) Re-parameterize f (x,y) as f (x(t),y(t))
3) Change ds using the arc length formula (which we learned previously)
4) Compute the integral

We also got into the fact that there's a physics aspect to the use of line integrals. (Sigh. Just when I thought I was done with physics)
Let's say you go a path on a 2-D xy coordinate plane with a vector field F (x,y). Think of the vector field as wind that you experience along your "path". We can use line integrals to figure out the net amount of force that is either helping or slowing you down along your "path". It's a cool beginning to the further expansion of line integrals.

     
Note:
The graph of the parametric curve where x (t) = sin (t)
                                                              y (t) = cos (t)
                                                              z (t) = t
 can be described in mutliple ways: a helix, a spring, a slinky, and the list continues on...