How do I calculate work with a non-constant force in a physics problem?

[Justin22]

I'm really stuck trying to figure out how to calculate the work done by a non-constant force for my physics assignment. My textbook gives the integral of force over displacement, but when I try to apply it to a specific problem where the force varies with position, I keep getting a different answer than the solution key.

[Ronald53]

Been there. I treated it as W = ∫ F(x) dx from x0 to x1, but when F(x) was something like a linear or quadratic function I kept getting numbers that didn’t match the key. I found it helped to double-check the limits and the sign: the work done by the force is the integral of F along the actual displacement, so in 1D it’s just F(x) dx with the path direction correct. I once accidentally used the final x as the lower limit and that wrecked the result. Another pitfall was confusing the variable in the function with the integration variable.

[MadisonAA]

I discretized it. split into many tiny dx, computed F at each x, summed F(xi) Δx. If the answer was off, I noticed I used x1 - x0 wrong or forgot that Δx should be signed along the path. Small mistakes there can flip the result.

[ZoeyG]

If F is given as a function of x, the exact form is ∫ F(x) dx between the start and end. For example if F = kx^2, W = (k/3)(x1^3 - x0^3). If you’re seeing a mismatch, maybe the problem isn’t just F(x) dx but F·dr along a curved path, or the force isn’t conservative and the path matters. The latter would explain a discrepancy with a simple endpoint check.

[John74]

I wandered into a tangent about energy diagrams, but I kept coming back to the same beat: you need the area under the F vs x curve between the start and end points. I’ve mixed up units near turning points before, which ruined the result. If you can share the exact F(x) and the x0, x1 values, I’ll try to spot where the mismatch hides.