Line Integral Calculator
Calculate line integrals of vector fields along parametric curves.
Vector Field F(x, y)
Parametric Curve r(t)
Parameter Range
Line Integral Formula
integral_C F dot dr = integral_a^b F(r(t)) dot r'(t) dt
Line Integral Result
0.000000
integral_C F dot dr
LArc Length
6.283185
WWork Done
0.000000
Curve Endpoints
Start Point(1.0000, 0.0000)
End Point(1.0000, -0.0000)
Curve Type
Closed Curve (start = end)
Properties
- If F = nabla f (conservative), integral depends only on endpoints
- For closed curves in conservative fields: integral = 0
- Work done = line integral of force field
- Arc length = integral of speed
About Line Integrals
Definition
A line integral integrates a function along a curve. For vector fields, it measures the work done by the field on a particle moving along the path.
Applications
- Work done by force fields
- Circulation in fluid dynamics
- Electric potential differences
- Mass of curved wires