How do I set up a line integral for work done by a variable force along a curve?

[RyanHM]

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 gives the general integral form, but when I try to set up the line integral for a specific example, I’m not sure if I’m correctly handling the dot product between the force vector and the displacement differential.

[Evelyn9]

I’ve been there. I try to write r(t) for the curve, then F(r(t)) and r'(t). Then the work along the path becomes ∫ F(r(t)) · r'(t) dt. The dot product is just F_x dx + F_y dy along the path.

[BrianG]

One time I mixed up the parameterization and ended with the wrong sign because the path was traversed the other way. It helps to plot the path and check the endpoints and direction.

[TimothyAH]

If you parametrize the curve with something simple like r(t) = (x(t), y(t)), you just do F_x(x(t),y(t)) x'(t) + F_y(x(t),y(t)) y'(t) and integrate dt. I know that sounds obvious, but I guess it’s easy to slip.

[KennethPA]

Sometimes I drift and start overthinking whether the curve is circular or something, and I lose track of the simple thing: the dot product uses the tangent to the path, not the straight line between endpoints. Then I try to re-check the parameterization and the integrand.