Line Integral Calculator

The Line Integral Calculator numerically evaluates a line integral along a parametric curve C: r(t) = (x(t), y(t)[, z(t)]), t ∈ [t0, t1] — either a scalar field integral ∫_C f ds (with respect to arc length) or a vector field integral ∫_C F · dr (with respect to displacement along the curve). Alongside the numeric result it reports the curve's arc length over the same interval and, for vector fields, a conservative-field check based on the curl of F.

It's built for vector calculus students checking a homework answer, and for anyone who needs a fast numeric line integral without setting up the substitution by hand.

How to use it

1. Choose the Integral type: scalar field (∫ f ds) or vector field (∫ F · dr).
2. Choose the Curve dimension: 2D or 3D.
3. Enter the curve's parametrization: x(t), y(t), and (in 3D) z(t) — standard math notation works: ^ for powers, * for multiplication, plus sqrt(), sin(), cos(), log(), and the other functions supported by mathjs's expression parser.
4. Enter the t lower bound (t0) and t upper bound (t1) — these accept constant expressions too, such as 2*pi, not just plain numbers.
5. For a scalar field, enter f(x, y[, z]). For a vector field, enter its components P(x, y[, z]), Q(x, y[, z]), and (in 3D) R(x, y, z).
6. Read the integral result (∫_C f ds or ∫_C F · dr), Arc Length, and — for vector fields — Conservative Field?. Step-by-Step Details shows dx/dt, dy/dt (and dz/dt in 3D), the integral setup, and the curl computation.

Results update automatically as you type. For example, the vector field F = (-y, x) around the unit circle (cos(t), sin(t)), t ∈ [0, 2π], returns a line integral of 6.28319 and is flagged not conservative.

How is the integral computed?

dx/dt, dy/dt, and dz/dt are found symbolically by differentiating x(t), y(t), and z(t). The curve, its derivative, and the field are then evaluated at 1001 evenly spaced points across [t0, t1] and combined with composite Simpson's rule — for a scalar field, integrating f(r(t)) · |r′(t)| dt; for a vector field, integrating [P dx/dt + Q dy/dt (+ R dz/dt)] dt. The same grid and rule compute the arc length, ∫ |r′(t)| dt.

What is the conservative-field check?

For a vector field F = (P, Q[, R]), the calculator computes the curl of F symbolically (∂Q/∂x − ∂P/∂y in 2D; the full curl vector in 3D) and checks whether it simplifies identically to zero. If it does, F appears conservative — the line integral between two points doesn't depend on the path taken between them, only on the endpoints. This partial-derivative test is a necessary but not sufficient condition on a domain with a hole or singularity (e.g. the origin removed from the plane), so treat a "conservative" verdict as strong evidence, not an absolute guarantee, on such domains.

What functions can I enter?

Anything mathjs's expression evaluator understands: polynomials, trigonometric functions (sin, cos, tan), roots and logarithms (sqrt(), log()), and combinations of these. Use t as the curve's parameter and x, y, z as the field's variables. Angles are in radians by default — write pi for π.

To find the curl of a 3D vector field at a single point directly, see the Curl Calculator; for a double integral over a rectangular region instead of along a curve, see the Double Integral Calculator.

Privacy

This calculator runs entirely in your browser. Your curve, field, and bounds are never uploaded, logged, or stored — the computation happens locally on your device.