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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?

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up vote 1 down vote accepted

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.

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

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