How do you set up the line integral for work by a variable force along a curve?

[Justin.G]

I’m really stuck on how to approach this physics problem about calculating the work done by a variable force along a curved path. My textbook just gives the integral formula, but when I try to set up the line integral with the given vector field, I’m not sure if my parameterization of the curve is correct or if I’m combining the components wrong.

[George60]

I’ve waded through this kind of thing before. I tried parameterizing the curve with a vector function r(t), then billed it as F(r(t)) · r'(t) and integrated, but I kept mixing components and losing track of which part goes with which variable.

[Ava.S]

One time I sketched x(t) and y(t) from the given path, computed dr/dt, and then did the dot product with the field along the path, and I still ended up with a mess that didn’t simplify. It felt like a bookkeeping error more than math.

[AmeliaYJ]

Is the real snag that the problem isn’t actually about the line integral but about whether the curve parameterization matches the path described in the setup or if the field is given in a rotated frame?

[JerryEW]

I did try a different tack once, grabbing the components separately: I treated the work as integral of F_x dx plus F_y dy along the path, and I kept forgetting to replace dx and dy with dx/dt dt and so on. It felt like I was chasing a bug rather than solving it.