How can the derivative of arc length be anything except zero?

Question

If a and b are constants, then the above definite integral (arc length) has to be some constant.

How can the derivative of s be anything except zero (this is contradicted in the blue box)? This method of evaluation is for line integrals by the way.

Answer 1

Pick a fixed starting time $t_0$, then you may define the arc length travelled after a time $t$ and denote it by $s(t)$. This is not constant in $t$, as you travel along the curve, the travelled distance increases. So $\frac{\mathrm ds}{\mathrm dt}$ is the velocity at a certain time.

Answer 2

The expression you highlighted gives the relation between $\mathrm ds$ and $\mathrm dt$, and a nonzero relationship should exist, as you would expect a correlation between the differential and change in $t$. In fact, it is equivalent to the integral you talked about above.