Does the starting/end point matter in line intergral of vector fields over a closed curve

Question

Suppose I have a vector field $F = (F_1, F_2):\mathbb{R}^2 \to \mathbb{R}^2$ and $C$ a positively oriented smooth simple closed curve. Then I know the definition of $$ \int_C F_1 dx + F_2 dy = \int_{a}^b F(r(t)).r'(t) dt $$ if say $C$ is parametrized by $r(t)$, $a \leq t \leq b$. Since $C$ is a closed curve I can also parametrize with the same function but starting at a different starting/end points, say $a + d \leq t \leq b + d$. For example, if $C = \{(\cos t, \sin t): 0 \leq t \leq 2 \pi\}$. Does changing start/end points matter when defining a line integral of a vector field over a closed curve? thank you