Arc Length Calculator

Formula for Arc Length

For a smooth curve given by y = f(x) on the interval [a, b], the arc length L is defined by the integral

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

In words: you square the derivative of f(x), add 1, take the square root, and integrate that quantity from x = a to x = b. The result is the distance along the curve between those two x-values.

The same relationship can 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 comes from applying the Pythagorean theorem to infinitesimally small segments of the curve.

How This Calculator Approximates Arc Length

Many arc length integrals cannot be expressed in a simple closed form. To handle such cases, this calculator uses numerical integration, specifically Simpson’s rule, to approximate the integral \int_a^b \sqrt{1 + (f'(x))^2} \, dx.

Key steps under the hood

• The interval [a, b] is divided into an even number of equal subintervals.
• The derivative f'(x) is computed symbolically using a math library.
• At each Simpson node x_i, the integrand \sqrt{1 + (f'(x_i))^2} is evaluated.
• Simpson’s rule combines these sample values with specific weights to approximate the integral and thus the arc length.

Simpson’s rule generally offers better accuracy than basic methods like the trapezoidal rule for smooth functions, while still being computationally efficient.

Interpreting the Result

The output of the calculator is a single numerical value representing the approximate length of the curve y = f(x) from x = a to x = b in the same units as your x- and y-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 the tool uses numerical integration with a fixed number of subintervals, the result is an approximation. For smooth, moderately curved functions, this approximation is usually very close to the exact arc length. For functions with steep slopes or rapidly changing curvature, the approximation error can be larger.

Worked Example: Parabola on a Short Interval

Consider 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 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 numerically. Instead of integrating by hand, we let the calculator approximate this integral using Simpson’s rule with a 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).

To illustrate how the integrand behaves at sample points, consider a small set of nodes:

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

Comparison: Exact vs. Numerical Perspectives

Different approaches can be used to compute or approximate arc length. The table below compares the main perspectives relevant to this calculator.

Assumptions and Limitations

To use this calculator effectively, it is important to understand the assumptions behind the arc length formula and the numerical method.

• 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 you suspect that your function or interval violates any of these assumptions, consider simplifying the function, narrowing the range, or splitting the interval into segments that each satisfy the conditions more clearly.

Practical Tips for Using the Calculator

• 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.