I was looking through my multivariable calculus homework and I saw an example where we needed to find the arclength of a simple space curve. It's very simple:

$$r(t)= cos(t)\hat i + sin(t)\hat j + t\hat k $$

Is the path defined by this curve not the same as the diagonal of the rectangle surface of a cylinder?

The formal length of the curve is given by $\int|r'(t)|dt$ and here for one complete revolution of the curve, the limits of this integral would be from 0 to 2$\pi$.

Doing the math results in a length of $2\pi\sqrt2$.

Using my presumed geometric method, I use the Pythagorean formula and plug in the circumference of the circle and height, but end up with a sum under the radical. In this case, I would plug in $2\pi$ and 1, because at $t=2\pi$, the point is $(1,0,2\pi)$ and at $t=0$, the point is $(1,0,0)$.

In other words, if I drew a diagonal across a rectangle $2\pi$ by $1$, and then curled the surface so it would form a topless & bottomless cylinder, isn't the line defined on the paper the same as the curve defined by $r(t)$?

I'm pretty sure my geometric assumption is wrong, but I'm stumped as to why. Could anyone provide an explanation as to why this is not the case?


Are you sure the height is 1? Check that again... (...$+t\hat k$), but your concept is correct =)

Unless you are saying your height is $2\pi$ and your circumference is $1$. In that case check your circumference. The diameter of the circle that makes it is 2.

  • $\begingroup$ Gosh darn it! You got me. I made a blunder. :D The height is $2\pi$ and the circumference is $2\pi$ $\endgroup$ – Zchpyvr Nov 1 '12 at 0:02

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