Part 8: Arc Length in Space

Arc length in space works in much the same way as arc length in the plane. For a parameterized space curve s(t) = (x(t),y(t),z(t)), we approximate a small piece of the curve by . Letting as before, we see that ds is really a vector tangent to the curve in space, namely

We can then compute the length of the curve by integrating the length |ds| along s(t) to get

1. Compute the arc length of the helix s(t) = (cos(t), sin(t), t) for values of t from 0 to . Is this a calculation that you could do without help from a computer? Why or why not? Plot the curve in your worksheet to get a visual sense of the length you have calculated.
2. Compute the arc length of the curve defined by s(t) = (cos(t), sin(t), sin(t)). Is the result different from your answer to step 1? Why or why not? Plot this curve with the helix from step 1 to see whether you would expect their lengths to be the same.
3. Compute the arc length of the curve s(t) = (cos(t²), sin(t²), t³) for values of t from 0 to . Is this a calculation that you could do without help from a computer? Why or why not? Again, plot the curve to get a visual sense of the length you have calculated.

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Copyright CCP and the author(s), 2001-2002