line integrals calculator - Aaron Graves, PhDude Replica

Interactive Line Integrals Calculator

Compute scalar line integrals and vector-field work integrals along a parameterized curve in 2D or 3D.

Integral type

Curve parameterization r(t)

Use standard math syntax. Example: cos(t), t^2, exp(-t). Constant pi is supported.

x(t)

y(t)

z(t) (optional)

t start

t end

Subintervals (even preferred)

Scalar form: ∫_C f(x,y,z) ds, where ds = |r'(t)|dt

f(x,y,z)

What this line integrals calculator does

This tool numerically evaluates line integrals along a user-defined parameterized curve. You can use it for two common tasks in vector calculus:

Scalar line integral of a function over arc length, useful for mass, charge, and weighted curve totals.
Vector line integral (work integral), useful for force fields and circulation.

1) Scalar line integral: ∫_C f ds

If your curve is r(t) = (x(t), y(t), z(t)) for t ∈ [a,b], then:

∫_C f ds = ∫_a^b f(x(t),y(t),z(t)) · |r'(t)| dt

This appears in applications such as computing total mass of a wire with position-dependent density.

2) Vector line integral: ∫_C F · dr

For vector field F = (P,Q,R) and curve r(t):

∫_C F · dr = ∫_a^b [P(x,y,z)x'(t)+Q(x,y,z)y'(t)+R(x,y,z)z'(t)] dt

This is the standard work integral in physics when a force field acts along a path.

How to use the calculator

Select the integral type from the dropdown.
Enter the curve components x(t), y(t), and optionally z(t).
Set t start and end values. You can use constants like pi.
Enter the field expression(s): either f(x,y,z) or P,Q,R.
Click Calculate.

Use explicit multiplication like 2*x rather than 2x.

Quick examples to try

Circle circulation

Let F = (-y, x, 0) and r(t) = (cos t, sin t, 0), t: 0 → 2*pi. This gives approximately 2*pi, a classic circulation result.

Scalar integral on a parabola

Use f(x,y,z)=x+y and r(t)=(t,t^2,0), t: 0 → 1. The tool evaluates the weighted arc-length accumulation numerically.

Common input mistakes

Using ^ is supported, but ensure parentheses are clear (e.g., (x+y)^2).
If your expression has domain limits (like sqrt(x)), make sure your path stays valid.
Increase subintervals for curves with sharp behavior or oscillation.

Why numerical integration?

Many line integrals do not simplify cleanly in closed form. Numerical methods provide fast, practical approximations and are ideal for engineering and applied math workflows.