What is a line integral of a scalar function?

A scalar line integral ∫_C f(x,y) ds sums the values of a function f weighted by infinitesimal arc length ds along a curve C. It differs from a vector line integral (work) — here the integrand is a plain function, not a dot product with a vector field.

How accurate is the Gaussian quadrature method?

The 16-point Gauss-Legendre rule is exact for polynomials up to degree 31 and converges rapidly for smooth integrands. For most textbook curves the error is below 10^-10. Highly oscillatory or discontinuous integrands may need a larger node count.

What syntax can I use in the expression fields?

Use standard math operators: +, -, *, /, ^ for power. Functions: sin, cos, tan, sqrt, abs, exp, log (natural log), log2, log10, PI, E. Example: x^2 * sin(y) + sqrt(t) in the curve fields where t is the parameter.

Can I use t directly in f(x,y)?

f must be written in terms of x and y, not t directly. The calculator substitutes x(t) and y(t) before evaluating, so use x and y as variable names in the integrand.

What does the step-by-step table show?

Each row shows one Gaussian node: the t value, x(t), y(t), f evaluated at that point, the arc-speed ds/dt, the product f·ds/dt, and its weighted contribution. The sum of contributions equals the integral.

Can I compute arc length with this tool?

Yes — set f(x,y) = 1. The line integral ∫_C 1 ds equals the arc length of the curve from t = a to t = b.

Line Integral Calculator

Name: Line Integral Calculator
Availability: InStock
Author: Nham Vu

Enter f(x,y), parametric equations x(t) and y(t), and the t-interval to compute the line integral numerically with a step-by-step breakdown.

Integral Setup

f(x, y) — integrand

Operators: + − * / ^ | sin, cos, tan, sqrt, exp, log, abs, PI

x(t) — x component

y(t) — y component

a (t start)

b (t end)

Quick examples

Result

Fill in the fields and click Calculate.

Step-by-step (Gauss nodes)

#	t	x(t)	y(t)	f(x,y)	ds/dt	w·f·ds/dt
Total

Copied!

Summary

Enter f(x,y), parametric equations x(t) and y(t), and the t-interval to compute the line integral numerically with a step-by-step breakdown.

How it works

Enter the integrand f(x,y) — any expression in x and y (e.g. x^2 + y, sqrt(x*y)).
Enter the parametric equations x(t) and y(t) that define the path.
Set the parameter interval [a, b] over which the curve is traced.
The calculator maps 16 Gaussian quadrature nodes from [a, b], evaluates x(t), y(t), dx/dt, and dy/dt at each node.
It computes ds/dt = sqrt((dx/dt)^2 + (dy/dt)^2) and multiplies by f(x(t), y(t)).
The weighted sum of those products gives the line integral ∫_C f(x,y) ds.

Use cases

Calculate the mass of a wire with variable density along a curved path.
Find arc length by setting f(x,y) = 1 and checking the result.
Verify hand-computed line integrals on homework or exam problems.
Explore how the path shape affects the value of a path-dependent integral.
Compute work done by a conservative scalar field along a curve.
Test parametric curve representations before doing symbolic integration.

Frequently Asked Questions

Last updated: 2026-07-23 · Reviewed by Nham Vu