Line Integral Calculator
Calculate Work: ∫C F · dr over a parametric curve.
1. Vector Field F(x, y)
2. Curve C: r(t)
Result (Work Done)
Path Visualization
Visual: Gray arrows show the force field. The Red line is the path. If the path goes with the arrows, work is positive.
Understanding Line Integrals
A normal integral adds up area under a curve. A Line Integral adds up a value as you travel along a specific curvy path through space.
The Physics: Work
If F represents a force (like wind or magnetism), the line integral calculates the total Work done by that force on a particle moving along the curve C.
Work = ∫ F · dr
The Math: How to Solve
We convert everything into terms of time t:
- Find velocity: r'(t) = <x'(t), y'(t)>
- Take Dot Product: F(r(t)) · r'(t)
- Integrate from tstart to tend.
Scalar vs. Vector Line Integrals
There are two types. A Scalar line integral (∫ f ds) measures something like the mass of a curved wire where density changes. A Vector line integral (∫ F · dr), which this tool calculates, measures flow or work along a path.
Frequently Asked Questions (FAQ)
Q: What if the answer is 0?
A: This means the net work is zero. It usually happens if the force is perpendicular to your movement everywhere, or if positive work cancels out negative work exactly.
Q: What is a “Closed Loop”?
A: A path that starts and ends at the same point (like a circle). If the field is “Conservative” (like gravity), the work done around any closed loop is always zero.
Q: Does speed matter?
A: No. The parameterization r(t) describes position over time, but the total work depends only on the path taken, not how fast you travel it (re-parameterization invariance).