Line Integral Calculator

Formula used

A line integral depends on both a field and a path. The calculator supports two common forms:

• Scalar field: ∫_C f(x,y,z) ds becomes ∫_a^b f(r(t)) · |r′(t)| dt.
• Vector field: ∫_C F·dr becomes ∫_a^b F(r(t)) · r′(t) dt.

For 2D, set z = 0 and the calculator uses only (x,y).

How to use this calculator

1. Select Scalar or Vector integral type.
2. Pick 2D or 3D (circle is always 2D).
3. Choose a curve: Parametric, Segment, or Circle.
4. Enter the field expression(s) and the curve definition.
5. Set the parameter interval a to b and steps.
6. Press Calculate. Use Download CSV/PDF to export.

Example data table

Line Integral Guide

1) What this calculator computes

This tool evaluates line integrals for scalar fields (∫C f ds) and vector fields (∫C F·dr). You define a curve and a field, and the calculator returns a single numeric value for the chosen path and interval. For vector fields, it integrates Fx·dx/dt + Fy·dy/dt + Fz·dz/dt along the parameter.

2) Supported curve models

You can enter a custom parametric curve r(t) with x(t), y(t), and optional z(t). For quick setup, choose a straight segment between two points or a 2D circle with center and radius. Circle uses an angle interval in radians, so 0 to 2*pi traces a full revolution.

3) Numeric method and accuracy

The integral is approximated with Simpson’s rule, which converges fast for smooth integrands. With step size h, error often decreases on the order of h⁴. The curve derivative r′(t) is estimated numerically using a small dt about h/10, with forward or backward differences near endpoints.

4) Step recommendations

Practical ranges are 200–2000 steps for most classroom examples, and 2000–8000 for sharper behavior. Steps are forced even (a Simpson requirement). Very large steps can slow down evaluation because expressions are computed at many sample points. If you suspect cancellation, compare runs at 400, 800, and 1600 steps and check that digits stabilize.

5) Expression features

Expressions may use variables x, y, z, t, constants pi and e, and functions like sin, cos, sqrt, abs, exp, and log. Power can be written as x^2 or x**2. Unknown identifiers are blocked to avoid unsafe evaluation, and multiplication should be explicit, like 2*pi.

6) Data checks with examples

The example table helps validate settings. For instance, f=1 around a circle of radius 2 should return circumference 4π ≈ 12.566. For a segment from (0,0) to (1,1) with f=x+y, the result is √2 ≈ 1.414. For a vector field F=(y,x) on the same segment, the value is 1.

7) Exporting results

After calculation, export a CSV summary for spreadsheets or a simple PDF report for sharing. Exports include integral type, dimension, curve choice, interval, steps, formula form, and the computed result, so you can reproduce and audit your work later. This is useful for notebooks, homework submissions, and comparison between paths.

FAQs

1) What is the difference between ∫C f ds and ∫C F·dr?

∫C f ds scales a scalar field by arc length, so speed |r′(t)| matters. ∫C F·dr projects a vector field onto the direction of motion using the dot product with r′(t).

2) Why does the calculator require an even number of steps?

Simpson’s rule works on pairs of subintervals and uses alternating weights (4 and 2). An even step count ensures the interval splits into an even number of subintervals, matching the method’s formula.

3) When should I increase the step count?

Increase steps if your result changes noticeably when recalculating, or if the curve has sharp turns, high frequency trig terms, or steep exponentials. More steps sample the integrand more densely and usually improve stability.

4) Can I compute in 3D for a parametric path?

Yes. Select 3D and provide z(t) for the curve. For scalar integrals, z is included in f(x,y,z). For vector integrals, the Fz component and dz/dt contribute to F·r′.

5) How do I set the circle interval?

Use theta_min and theta_max in radians. For a full circle, set 0 to 2*pi. For a quarter circle, use 0 to pi/2. These values override the general a and b inputs.

6) Why do I get “invalid characters” or “unknown identifier” errors?

The parser only allows a limited set of variables and math functions. Remove extra letters, unsupported symbols, or custom function names. If you need a constant, use pi or e, and write multiplication explicitly, like 2*pi.