Line Integral Calculator
Calculate line integrals of scalar fields (∫f ds) and vector fields (∫F·dr) along parametric curves in 2D and 3D. Enter the field, parametric equations, and bounds to get symbolic results with step-by-step solutions, arc length, and an interactive 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.
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
pior2*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.
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"Line Integral Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-08
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