One of my tutors used a notation like this:
$ \int_{t=a}^{t=b} F(r)$
and kept this notation later when he had written it as
$ \int_{t=a}^{t=b} f_s(s) ds$
and later had it written as an integral in terms of $dt$ where it was easy to use the subsitution
The earlier:
$ \int_{t=a}^{t=b} F(r)$
notation seems wrong, was he just doing this to make it clear we want to have limits to substitute for $t$? and in fact to have the notation:
$ \int_{C}^{} F(r)$ is the correct way, changing the limits to $a$ and $b$ when we have an integrand in terms of $t$?
Please don't use that notation. It should be either $\int_CF(r)\cdot dr$ if you're calculating line integrals (maybe you want to use bold face to emphasize the vector and dot product nature), or just $\int_C\left(\sum_{i=1}^nF_i\,dx^i\right)$. Or, if you're doing integrals with respect to arclength, something like $\int_Cf(s)\,ds$. In any case, you should always write $\int_C$ to show you're integrating over the curve $C$. Only afterwards (say after you parametrize $C$ say via some $\gamma:[a,b]\to\Bbb{R}^n$) should you have $\int_a^b(\text{stuff})\,dt$.
The reason for this is that line integrals (of $1$-forms) i.e $\int_C\sum_{i=1}^nF_i\,dx^i$ typically depend on the curve chosen. It's not just about the endpoints. If by some miracle the line integral happens to only depend on endpoints of the curve, then that is a very special situation (conservativeness). So, writing $\int_{t=a}^{t=b}F(r)\,dr$ or something like this completely suppresses the curve in the notation, and hence is conceptually misleading (and also notationally wrong).