Line Integral Calculator

Calculate line integrals of scalar and vector fields along parametric curves in 2D and 3D. Enter the field, parametric equations and bounds for symbolic results with step-by-step solutions, arc length and a curve visualization.

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About Line Integral Calculator

The Line Integral Calculator evaluates both scalar line integrals \(\int_C f\,ds\) and vector line integrals \(\int_C \mathbf{F} \cdot d\mathbf{r}\) along parametric curves in 2D and 3D space. Enter the field, parametric equations, and parameter bounds to get a full step-by-step solution with symbolic results, arc length computation, and an animated curve visualization.

∫ Scalar Line Integral Integrates f along the curve weighted by arc length ds

⟳ Vector Line Integral Dot product of F with tangent dr measures work

📏 Arc Length Total length of the curve from r(a) to r(b)

🔄 Conservative Fields Path-independent integrals when ∇×F = 0

Line Integral Formulas

Type	Formula	Description
Scalar ∫f ds	\(\int_C f\,ds = \int_a^b f(\mathbf{r}(t))\,\|\mathbf{r}'(t)\|\,dt\)	Integrate scalar function along curve weighted by speed
Vector ∫F·dr	\(\int_C \mathbf{F}\cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt\)	Dot product integral measures work or circulation
Arc Length	\(L = \int_a^b \|\mathbf{r}'(t)\|\,dt\)	Total length of the parametric curve
Conservative	\(\int_C \nabla\phi\cdot d\mathbf{r} = \phi(\mathbf{r}(b)) - \phi(\mathbf{r}(a))\)	Fundamental Theorem of Line Integrals

How to Use the Line Integral Calculator

Choose the integral type. Select "∫f ds" for a scalar line integral or "∫F·dr" for a vector (work/circulation) line integral.
Select the dimension. Choose 2D or 3D depending on your curve and field.
Enter the field. For scalar integrals, type the function f(x, y) or f(x, y, z). For vector integrals, enter each component P, Q, and R.
Define the parametric curve. Enter x(t), y(t), and optionally z(t). Use standard math notation — cos(t), t^2, sin(t), etc.
Set the bounds. Enter the start and end values of t. You can use expressions like pi or 2*pi.
Click Calculate to see the step-by-step solution, numeric result, arc length, and curve animation.

Common Parametric Curves

Curve	Parametrization	Bounds
Circle (radius R)	x = R cos(t), y = R sin(t)	t ∈ [0, 2π]
Line segment A→B	r(t) = (1−t)A + tB	t ∈ [0, 1]
Parabola y = x²	x = t, y = t²	t ∈ [a, b]
Helix	x = cos(t), y = sin(t), z = t	t ∈ [0, 2π]
Ellipse	x = a cos(t), y = b sin(t)	t ∈ [0, 2π]

Understanding the Results

The calculator provides several pieces of information in the result:

Integral Value: The exact symbolic result (when possible) and its numerical approximation.
Arc Length: The total length of the curve, computed as \(\int_a^b |\mathbf{r}'(t)|\,dt\).
Speed |r'(t)|: The magnitude of the velocity vector, which acts as the arc length element.
Conservative Field Check: For vector integrals, the calculator checks if ∇×F = 0 (the field is conservative). Conservative fields have path-independent integrals.
Curve Visualization: An animated plot of the parametric curve showing direction of traversal with a moving dot tracing the path.

Frequently Asked Questions

What is a line integral?

A line integral computes the integral of a function along a curve. For scalar fields, it sums up the values of f weighted by arc length (∫f ds). For vector fields, it sums the component of F along the tangent direction (∫F·dr), often interpreted as work done by a force field.

What is the difference between a scalar line integral and a vector line integral?

A scalar line integral ∫_C f ds integrates a scalar function f along a curve weighted by the arc length element ds, giving the total accumulated value of f along the path. A vector line integral ∫_C F·dr integrates a vector field F along a curve by taking the dot product with the tangent vector dr, measuring how much F pushes along the curve direction. Scalar integrals are used for mass and average value problems; vector integrals compute work and circulation.

How do you parametrize a curve for a line integral?

A parametric curve r(t) expresses each coordinate as a function of a single parameter t. For example, a circle of radius R is parametrized as x(t) = R cos(t), y(t) = R sin(t) with t from 0 to 2π. The line integral formula then converts the curve integral into a standard definite integral over t.

When is a vector line integral path independent?

A vector line integral is path independent when the vector field F is conservative, meaning its curl is zero everywhere in a simply connected domain. In that case, F equals the gradient of a potential function φ, and the integral depends only on the values of φ at the endpoints, not on the specific path taken. The calculator automatically checks for this condition.

What is the physical meaning of a line integral?

Physically, a scalar line integral can represent the mass of a wire with varying density, or the total heat along a path. A vector line integral commonly represents the work done by a force field on a particle moving along the curve, or the circulation of a fluid velocity field around a loop. In electromagnetism, line integrals appear in Ampere's law and Faraday's law.

What math notation does the calculator accept?

Use standard math notation: ^ for exponents (x^2), * for multiplication (2*x, though implicit multiplication like 2x also works), and standard function names like sin, cos, tan, exp, log, sqrt. For the parameter bounds, you can enter expressions like pi, 2*pi, or numerical values.

Reference this content, page, or tool as:

"Line Integral Calculator" at https://MiniWebtool.com/line-integral-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-08

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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