Arc Length Calculator

Formula for the Arc Length of y = f(x)

For a smooth graph y = f(x) on the interval [a, b], this calculator’s arc length L is defined by the integral

L = \displaystyle \int_a^b \sqrt{1 + (f'(x))^2} \, dx

For this graph-based arc length calculation, square the derivative of f(x), add 1, take the square root, and integrate from x = a to x = b. The result is the distance traced along the curve between those x-values.

The arc length relationship can also be written with MathML notation:

Here, df/dx (or f'(x)) is the derivative of f(x) with respect to x. The square-root term follows from the Pythagorean theorem applied to infinitesimally small pieces of the graph.

Interpreting an Arc Length Result

An arc length result is one numerical estimate of the length of the graph y = f(x) from x = a to x = b, expressed in the compatible units of the axes.

• If x is measured in meters and f(x) outputs meters, then the reported arc length is in meters.
• If your input values are dimensionless (pure numbers), the result is a dimensionless length.
• A result slightly larger than the straight-line distance between (a, f(a)) and (b, f(b)) reflects the curvature of the function in that interval.

Because this arc length tool uses numerical integration with a fixed number of subintervals, its output is an approximation. For smooth, moderately curved functions, the estimate is often close to the exact arc length. Functions with steep slopes or rapidly changing curvature can produce larger numerical error.

Worked Example: Arc Length of a Parabola on a Short Interval

This arc length example uses the function f(x) = x^2 on the interval [0, 1].

1. Compute the derivative. For f(x) = x^2, we have f'(x) = 2x.
2. Set up the arc length integrand. The integrand becomes \sqrt{1 + (f'(x))^2} = \sqrt{1 + (2x)^2} = \sqrt{1 + 4x^2}.
3. Write the arc length integral. We want L = \int_0^1 \sqrt{1 + 4x^2} \, dx.
4. Approximate the curve length numerically. Instead of integrating by hand, let the calculator estimate this integral using Simpson’s rule with its fixed number of subintervals.

If you enter f(x) = x^2, a = 0, and b = 1 into the calculator, the result will be a value slightly larger than 1. The increase above 1 reflects the fact that the curve bends away from the straight line connecting (0, 0) and (1, 1).

The following nodes show how the parabola’s arc length integrand changes across the interval:

As x increases, the derivative f'(x) = 2x grows, making the integrand larger and contributing more to the parabola’s total arc length.

Arc-Length Methods: Exact and Numerical Approaches

Arc length can be found symbolically in some cases or estimated numerically when no convenient antiderivative exists. The table below compares approaches relevant to this calculator.

Arc-Length Assumptions and Limitations

This graph arc length calculation depends on the behavior of the entered function and on the fixed-step numerical method used to estimate its integral.

• Differentiable function on [a, b]. The formula L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx assumes that f(x) has a well-defined derivative on the entire interval [a, b]. If the function is not differentiable at some point, the integral may not represent a true arc length there.
• No discontinuities or vertical tangents. If f(x) has jumps, asymptotes, or vertical tangents between a and b, then f'(x) can become undefined or extremely large. In such regions, the numerical approximation may be unstable or misleading.
• Fixed number of Simpson subintervals. This implementation uses a fixed, finite number of subintervals for Simpson’s rule. For functions with gentle curvature, this usually produces a good approximation. For highly oscillatory or very steep functions, more refined methods (or more subintervals) may be necessary for high accuracy.
• Symbolic differentiation limits. The derivative is obtained through a symbolic or analytic differentiation routine. Certain exotic or poorly specified functions may not be parsed correctly, which can lead to errors or warnings instead of a meaningful result.
• Numerical round-off. As with any floating-point numerical computation, very large or very small values can introduce round-off error. For most practical ranges of a, b, and f(x), this effect is minor compared to the discretization error from Simpson’s rule.

If your arc length interval includes a problematic point, try narrowing the range or splitting the curve into segments whose derivatives and values are well behaved.

How to Use the Arc Length Calculator

To obtain a meaningful arc length estimate, enter a single-variable graph and an interval over which that graph can be differentiated and evaluated.

• Stick to one variable. Enter functions of x only (for example, sin(x), x^2 + 3x, exp(-x)). Multivariable expressions are not supported.
• Choose a and b where the function is smooth. Avoid intervals that cross discontinuities, corners, or cusps, as these can break the arc length formula.
• Check units. Use consistent units for x and f(x) so that the arc length result has a clear physical meaning.
• Sanity check results. Compare the arc length to the straight-line distance between the endpoints. The arc length should always be at least as large as that distance.

How This Calculator Approximates Arc Length

For arc length integrals without a simple closed form, this calculator uses Simpson’s rule to estimate the integral \int_a^b \sqrt{1 + (f'(x))^2} \, dx.

Simpson-Rule Steps for This Arc Length Estimate

• The interval [a, b] is divided into 10 equal subintervals.
• The derivative f'(x) is computed symbolically using a math library.
• At each of the 11 Simpson nodes x_i, the integrand \sqrt{1 + (f'(x_i))^2} is evaluated.
• Simpson’s rule combines these curve-length samples with specific weights to estimate the integral and thus the arc length.

For smooth functions, Simpson’s rule generally gives a more accurate arc length estimate than basic trapezoidal integration while remaining computationally efficient.