Line Integral Calculator
Integral Type
Scalar Function
Path C
Understanding Line Integrals
A line integral computes the integral of a function along a curve. There are two main types: scalar line integrals and vector line integrals.
Scalar Line Integral
∫C f(x,y) ds
Integrates a scalar function along a curve weighted by arc length
Vector Line Integral
∫C F·dr
Computes work done by a force field along a path
Computing Line Integrals
Scalar Line Integral
∫C f ds = ∫ab f(x(t), y(t)) · |r'(t)| dt
where |r'(t)| = √[(dx/dt)² + (dy/dt)²]
Vector Line Integral
∫C F·dr = ∫ab [P·(dx/dt) + Q·(dy/dt)] dt
Conservative Vector Fields
A vector field F = (P, Q) is conservative if ∂Q/∂x = ∂P/∂y. For conservative fields:
- The line integral is path-independent
- ∫C F·dr = φ(end) - φ(start) where F = ∇φ
- Line integral around any closed curve is zero
Applications
Physics
- • Work done by a force
- • Circulation of a fluid
- • Mass of a wire
Geometry
- • Arc length of curves
- • Center of mass
- • Moments of inertia
About Line Integral Calculator - Scalar & Vector Fields
Calculate line integrals for scalar and vector fields along parametric curves. Compute work, circulation, and arc length integrals.
Our **Line Integral Calculator** computes integrals along curves for both scalar fields (∫f ds) and vector fields (∫F·dr). Enter your function, parametric path, and bounds to get numerical results with step-by-step solutions. For convergence analysis, see our Interval of Convergence Calculator.
Scalar line integrals weight a function by arc length, useful for mass of wires and surface area. Vector line integrals compute work done by forces along paths—fundamental in physics and engineering.
Supports common paths (lines, circles, parabolas) and custom parametric curves. The calculator checks if vector fields are conservative (path-independent) and evaluates using numerical integration.
Key Features
Why Use This Tool?
Common Use Cases
Physics: Work done by force along a path.
Fluid Mechanics: Circulation around a curve.
Geometry: Arc length and weighted averages.
Electromagnetism: Field integration.
Related Tools
How to Use
Choose scalar or vector integral
Enter function or field components
Select path type and parameters
Enter parameter bounds
Click Calculate