Helix's arc length

Question

I'm reading this.

The relevant definitions are that of parametrized curve which is at the beginning of page 1 and the definition of arclength of a curve, which is in the first half of page 6.

Also the author mentions the helix at the bottom of page 3.

On exercise $1.1.2.$ (page 8) I'm asked to find the arc length of the helix:

$\alpha (t)=(a\cos (t), a\sin (t), bt)$, but the author don't say what the domain of $\alpha$ is.

How am I supposed to go about this?

Usually when the domain isn't specified isn't the reader supposed to assume the domain is a maximal set? In that case the domain would be $\Bbb R$ and the arc length wouldn't be defined as the integral wouldn't be finite.

Answer 1

It seems sensible to do it for one complete cycle of sine and cosine, that is, any interval of length $2\pi$. So we are measuring the length of one complete turn around the cylinder that the helical vine climbs on.

Answer 2

There are a number of ways of approaching this problem. And yes, you are correct, without the domain specified there is a dilemma here. You can give an answer for one complete cycle of $2\pi$. Depending on the context you may find it more convenient to measure arc length as a function of $z$-axis distance along the helix... a sort of ratio: units of length along the arc per units of length of elevation. Thirdly, you can also write the arc length not as a numeric answer but as a function of $a$ and $b$ marking the endpoints of any arbitrary domain. Personally, I recommend doing the third and last. Expressing the answer as a function is the best you can do without making assumptions about the domain in question, and it leaves a solution that can be applied and reused whenever endpoints are given.