Study work and circulation on parameterized curves. Compare integrands, inspect values, and download polished summaries. Build intuition through visuals, formulas, examples, and practical exports.
Enter your parameterized path and field expressions. The result, graph, and export controls will appear here after submission.
This example uses the default vector setup: F = (y, -x, 0) and r(t) = (cos t, sin t, 0), for 0 ≤ t ≤ 2π.
| t | x(t) | y(t) | dx/dt | dy/dt | Integrand |
|---|---|---|---|---|---|
| 0.000 | 1.000 | 0.000 | 0.000 | 1.000 | -1.000 |
| 1.571 | 0.000 | 1.000 | -1.000 | 0.000 | -1.000 |
| 3.142 | -1.000 | 0.000 | 0.000 | -1.000 | -1.000 |
| 4.712 | 0.000 | -1.000 | 1.000 | 0.000 | -1.000 |
Because the integrand stays near -1 around the circle, the integral is close to -2π.
Vector line integral: ∫ab F(r(t)) · r′(t) dt
For F = (P, Q, R) and r(t) = (x(t), y(t), z(t)), the calculator evaluates P dx/dt + Q dy/dt + R dz/dt.
Scalar line integral: ∫C f ds = ∫ab f(r(t)) |r′(t)| dt
Here |r′(t)| = √((dx/dt)² + (dy/dt)² + (dz/dt)²), which converts a parameter change into arc-length change.
Numerical method: Composite Simpson's Rule
The page samples the integrand across evenly spaced parameter values, estimates derivatives numerically, and integrates the resulting function accurately.
Choose the integral mode first. Use vector mode for work or circulation, and scalar mode for density or mass along a curve.
Enter x(t), y(t), and z(t). For a planar curve, keep z(t) equal to 0.
In vector mode, fill P, Q, and R. In scalar mode, fill the scalar field expression f(x,y,z).
Set the parameter start and end values. Increase intervals for a smoother result when expressions change rapidly.
Submit the form. Review the result cards, the curve graph, the integrand graph, and the sampled table.
Use the export buttons to save the sample table as CSV or create a PDF summary for reporting or revision.
It computes parametric line integrals along a user-defined curve. You can evaluate either a vector field dot product with the path differential or a scalar field accumulated over arc length.
Use vector mode when you need work, circulation, or signed accumulation from a vector field. The calculator evaluates F(r(t)) · r′(t) across the chosen parameter interval.
Use scalar mode when a scalar field is distributed along a path, such as density, temperature, or concentration. The calculator multiplies the field by arc-length speed before integrating.
Yes. Enter x(t), y(t), and z(t). If your path is only two-dimensional, simply leave z(t) as 0 and the calculator still works correctly.
The page uses numerical integration, so more intervals usually improve accuracy. Increase the interval count when the curve bends sharply or the field changes quickly.
You can use t, x, y, z, pi, e, and standard functions such as sin, cos, tan, log, sqrt, abs, min, max, and pow.
One graph displays the parameterized curve in two or three dimensions. The second graph shows how the integrand changes with the parameter, which helps explain the final total.
You can export the sampled calculation table as CSV for spreadsheets or generate a PDF summary containing the main results and visible sample rows.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.