Line Integral Calculator

Work out line integral instantly with clear inputs, formula shown and shareable results.

About Line Integral Calculator

For a constant field the line integral reduces to the dot product F·d, independent of the path taken. That path independence is the hallmark of a conservative field.

Formula

Line integral

∫C F · dr = F · d for a constant field over displacement d

Worked example

With the default values below, Line Integral Calculator returns Line integral (work done) of 50.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Constant field component Fx: 3
Constant field component Fy: 4
Path start (x, y): 0, 0
Path end (x, y): 6, 8

Results

Line integral (work done): 50.00000000
Straight-line path length: 10.00000000
Field magnitude: 5.00000000
cos of the angle between field and path: 1.00000000

Step by step

1Values used
2Line integral
3Line integral (work done)= 50.00000000
4Straight-line path length= 10.00000000
5Field magnitude= 5.00000000
6cos of the angle between field and path= 1.00000000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For F = (3, 4) from the origin to (6, 8)?

The work is 18 + 32 = 50, and since the field and path are parallel the cosine is exactly 1.

When does the path matter?

Whenever the field has non-zero curl. Then different routes between the same endpoints give different integrals.