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?