Are regular/Riemann integrals a special case of a line integral?

Classical definition, taking for example Tom M. Apostol, Calculus, Volume II_ Multi-Variable Calculus and Linear Algebra, define line integral for piecewise smooth path $alpha$ defined in $n$ space on interval $[a,b]$ with $C$ as graph of $alpha$ and vector field $F$, defined on $C$, by equation $$intlimits_{C} F d alpha = intlimits_{a}^{b}F(alpha(t))cdot alpha^{'}(t)dt$$ whenever integral on right hand exists.

Now, if we take, $alpha :[t_0,t_1] to mathbb{R}^2$, defined as $alpha(t)=(t,0)$, $F=(f,0)$ as field defined on graph of $alpha$, then we will have $$intlimits_{C} F d alpha =intlimits_{a}^{b}f(t)dt$$ which, as I guess, you wanted to express.

Actually, you're integrating $f(x)$, not $f(x,y)$. You don't need to set the curve inside $Bbb R^2$ (or $Bbb R^3$). Line integrals make sense on "curves" in $Bbb R$, the simplest case being the directed line segment $[a,b]$. You can indeed think of $displaystyleint_a^b f(x),dx$ as the work done by the force field $vec F = f(x)vec i$ moving a test object along the interval. (Indeed, the spring compression problems and such that appear in standard Calc II classes are examples of this.)