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Arc Length of a Curve Calculator

Find the arc length of a curve y = f(x) from x = a to x = b with L = ∫√(1 + f′(x)²) dx, plus the straight-line distance between the endpoints and how much longer the curve is.

f(x)

From x = a

To x = b

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How it works

L = ∫ₐᵇ √(1 + [f′(x)]²) dx · f′ by central differences, integral by Simpson’s rule with 4,000 panels · chord = √((b − a)² + (f(b) − f(a))²)

Arc length adds up tiny hypotenuses along the curve: over a small step dx the curve rises f′(x)·dx, so the piece of curve has length √(1 + f′(x)²)·dx. Integrating from a to b gives the total length (OpenStax Calculus Vol. 1 §6.4). Most arc-length integrals have no elementary antiderivative, so the calculator evaluates it numerically with a central-difference derivative and Simpson’s rule, accurate to about six decimals for smooth curves.

Worked example

y = x² from x = 0 to x = 1: f′(x) = 2x, so L = ∫₀¹ √(1 + 4x²) dx = √5/2 + ln(2 + √5)/4 ≈ 1.118034 + 0.360909 = 1.478943. The straight line from (0, 0) to (1, 1) is √2 = 1.414214 long, so the curve is about 1.045771 times as long as the chord.

Sources: OpenStax Calculus Vol. 1 — 6.4 Arc Length of a Curve and Surface Area · Paul’s Online Math Notes — Arc Length

Frequently asked questions

What is the arc length formula?

For y = f(x) on [a, b], L = ∫ from a to b of √(1 + (dy/dx)²) dx. For a parametric curve it is ∫ √((dx/dt)² + (dy/dt)²) dt.

Why is the arc length integral so hard to do by hand?

The square root of 1 + f′(x)² rarely simplifies, so only a few textbook curves (like y = x^(3/2) or cosh x) have neat answers. Numerical integration is the standard approach otherwise.

Is the arc length always longer than the straight line?

Yes, unless the curve is itself a straight segment, in which case they are equal. The ratio shown tells you how much the curve bends.

Does this calculate the arc length of a circle?

Use the circle arc length calculator for s = rθ. This tool is for the length of a graph y = f(x), such as a parabola or a sine wave.

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This calculator is for educational purposes. Double-check important results before acting on them.