Find the work done by the force field $F =\langle x^2 , ye^x \rangle $ to move a particle along the curve $x = y^2 + 1$, from $A = (1, 0)$ to $B = (2, 1)$.
Can I parameterise this curve as a line segment from point $A$ to point $B$?
$$ x=(1-t)x_0+tx_1 $$
$$ x= 1+t $$
$$ y = (1-t)y_0+t(y_1) $$
$$ y=1 $$
Or do I have to parameterise the function itself?
i.e.
$$y=t $$
$$ x= t^2 +1 $$
The parametrization
$$ x=(1-t)x_0+tx_1 $$
$$ x= 1+t $$
$$ y = (1-t)y_0+t(y_1) $$
$$ y=1 $$
parametrizes the straignt line from one point to the other, so yes, you need to parametrize like you did in the second example. That is the simplest of all parametrizations that are available.