Length of Curve Calculator: Formula & Steps

This calculator finds the exact length of a curve defined by a function y = f(x) between two x-values, using the standard arc length integral. Enter the function, the lower limit a, the upper limit b, and the number of subintervals for the numerical approximation; you get the approximate curve length in the same units as x and y. The page also explains the formula, shows a worked example, and covers common pitfalls.

Formula

The arc length L of a smooth curve y = f(x) from x = a to x = b is given by the integral:

L = ∫_a^b √(1 + [f′(x)]²) dx

Where:

• f(x) is the function defining the curve.
• f′(x) is the derivative of f(x) with respect to x.
• a and b are the x-coordinates of the endpoints (with a ≤ b).
• dx is the differential of x.
• The result L is in the same linear units as x and y (e.g., meters, feet).

For a curve given parametrically as x(t) and y(t) from t = t₁ to t = t₂, the formula becomes:

L = ∫_t1^t₂ √( (dx/dt)² + (dy/dt)² ) dt

For a polar curve r(θ) from θ = α to θ = β, the arc length is:

L = ∫_α^β √( r² + (dr/dθ)² ) dθ

Worked Example

Using the default values: f(x) = x², a = 0, b = 2, n = 100.

Step 1: Find the derivative f′(x) = 2x.

Step 2: Write the integrand: √(1 + (2x)²) = √(1 + 4x²).

Step 3: Set up the integral: L = ∫₀² √(1 + 4x²) dx.

Step 4: Evaluate numerically. Using Simpson’s rule with 100 subintervals gives:

L ≈ 4.6468 (rounded to 4 decimal places).

This matches the calculator’s output for the default values.

When to Use Arc Length

Arc length calculations appear in many fields:

• Geometry: finding the length of a curved boundary or a cable hanging between two points.
• Physics: computing the distance traveled by a particle along a known trajectory.
• Engineering: determining the length of a curved beam, pipe, or road segment.
• Computer graphics: measuring the length of a spline or Bezier curve for animation or rendering.

Use this calculator when you have a continuous, differentiable function and need the length of its graph between two points. For piecewise functions, break the curve at the piece boundaries and sum the lengths.

How the Numerical Method Works

Most arc length integrals cannot be evaluated with elementary antiderivatives. For example, ∫√(1 + x³) dx has no simple closed form. Therefore, calculators use numerical integration techniques:

• Simpson’s rule: approximates the integrand by quadratic polynomials over each pair of subintervals. It is accurate for smooth functions and converges quickly.
• Trapezoidal rule: uses linear segments; simpler but less accurate for the same number of intervals.
• Adaptive methods: refine the step size where the function changes rapidly.

The calculator uses Simpson’s rule with n subintervals (n must be even). The error decreases as n increases, but so does computation time. For typical curves, n = 100 to 1000 gives good accuracy.

Common Mistakes and Pitfalls

• Forgetting the derivative: Using f(x) instead of f′(x) in the integrand gives the wrong result.
• Unit mix-ups: The result is in the same units as x and y. If x is in meters and y is in meters, the length is in meters. If the axes have different scales (e.g., x in seconds, y in meters), the result is not a physical length but a mathematical one.
• Ignoring the square root: The integrand is √(1 + (f′)²), not 1 + (f′)².
• Reversing limits: Arc length is always positive; if a > b, the integral gives a negative value. The calculator assumes a ≤ b.
• Rounding too early: Round intermediate values only at the end to avoid large errors.

Limits of the Formula

The arc length formula assumes the curve is smooth (continuously differentiable) on [a, b]. If the curve has a corner or vertical tangent, the derivative may be undefined or infinite, and the integral may not converge. For example, the curve y = x^(2/3) has a cusp at x = 0; the arc length from -1 to 1 is finite, but the derivative is infinite at 0, so the formula must be handled carefully (split the interval).

Also, the formula gives the length of the graph of the function, not the distance along the x-axis. For a straight line segment, the formula reduces to the Euclidean distance between endpoints.

Alternatives and Related Calculations

• Distance between two points: For a straight line, use the distance formula √((x₂-x₁)² + (y₂-y₁)²).
• Length of a parametric curve: Use the parametric version of the formula.
• Length of a polar curve: Use the polar version.
• Surface area of revolution: Similar integral but with an extra factor of 2πy (or 2πx) for rotating about the x-axis (or y-axis).

If you have discrete data points instead of a function, you can approximate the curve length by summing the straight-line distances between consecutive points (polygonal approximation).

Frequently Asked Questions

Q: What is the arc length formula?
A: For y = f(x), the arc length from a to b is ∫√(1 + (f′(x))²) dx. For parametric curves, use ∫√((dx/dt)² + (dy/dt)²) dt.

Q: Can I use this calculator for any function?
A: It works for any function you can type in, as long as it is defined and differentiable on the interval. For functions with vertical tangents or discontinuities, the result may be inaccurate or undefined.

Q: How do I enter a function like sin(x) or e^x?
A: Use standard notation: sin(x), exp(x) or e^x, sqrt(x), etc. The calculator uses the same syntax as most graphing tools.

Q: Why is the result an approximation?
A: Most arc length integrals have no elementary antiderivative, so numerical methods (Simpson’s rule) are used. The approximation improves with more subintervals.

Q: What units does the result have?
A: The result is in the same units as the x and y coordinates. If both are in meters, the length is in meters; if x is in seconds and y in meters, the result is a mathematical length with mixed units, not a physical length.

Q: How do I find the length of a curve given by data points?
A: Sum the distances between consecutive points using the distance formula. For a smooth curve, you can also fit a polynomial or spline and then use the arc length integral.

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in f | Function f(x) | - | x^2 | 0
in a | Lower limit a | - | 0 | 0
in b | Upper limit b | - | 2 | 0
in n | Subintervals n | - | 100 | 1
out arc | Approx. arc length | units | simpson(f, a, b, n) | 4