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.
Your ad blocker is preventing us from showing ads
MiniWebtool is free because of ads. If this tool helped you, please support us by going Premium (ad‑free + faster tools), or allowlist MiniWebtool.com and reload.
- Allow ads for MiniWebtool.com, then reload
- Or upgrade to Premium (ad‑free)
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.
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
You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.
Related MiniWebtools:
Calculus:
- Convolution Calculator Featured
- Derivative Calculator
- Directional Derivative Calculator
- Double Integral Calculator
- Implicit Derivative Calculator
- Integral Calculator
- Inverse Laplace Transform Calculator
- Laplace Transform Calculator
- Limit Calculator
- Partial Derivative Calculator
- Single Variable Derivative Calculator
- Taylor Series Calculator
- Triple Integral Calculator
- Radius of Convergence Calculator
- Curvature Calculator
- Wronskian Calculator
- Runge-Kutta (RK4) Method Calculator
- Fourier Series Coefficients Calculator
- Volume of Revolution Calculator New
- Surface of Revolution Calculator New
- Riemann Sum Calculator New
- Trapezoidal Rule Calculator New
- Simpson's Rule Calculator New
- Improper Integral Calculator New
- L'Hôpital's Rule Calculator New
- Maclaurin Series Calculator New
- Power Series Calculator New
- Series Convergence Test Calculator New
- Infinite Series Sum Calculator New
- Average Rate of Change Calculator New
- Instantaneous Rate of Change Calculator New
- Related Rates Solver New
- Optimization Calculator (Calculus) New
- Gradient Calculator (Multivariable) New
- Divergence Calculator New
- Curl Calculator New
- Line Integral Calculator New
- Surface Integral Calculator New
- Newton's Method Calculator New
- First-Order ODE Solver New
- Second-Order ODE Solver New
- Direction Field / Slope Field Plotter New
- Euler's Method Calculator New
- Bernoulli ODE Solver New
- System of ODEs Solver New
- Fast Fourier Transform (FFT) Calculator New
- Z-Transform Calculator New
- Numerical Integration Calculator New