About this tool
The Arc Length Calculator is a free, in-browser tool that measures the true length of a curve y = f(x) between two x-values, following the winding path rather than the straight-line distance. It applies the standard formula L = ∫ from a to b of √(1 + [f′(x)]²) dx, which sums up infinitesimal segments √(dx² + dy²) along the curve. The derivative f′(x) is computed numerically, so any function of x you type works without you differentiating it.
Everything is calculated locally in your browser using a small expression parser, so nothing you enter is uploaded. Enter the function with x as the variable (for example x^2, sin(x), or x^(3/2)) and the interval endpoints a and b, and the tool integrates the arc-length integrand with composite Simpson's rule, doubling the number of subintervals until successive estimates agree to a tight tolerance for an accurate result. If you enter the endpoints in either order it uses the interval between them.
Arc length is a classic calculus II application for the distance travelled along a curve — the length of a road, wire, or graph. The result panel also shows the straight chord distance between the endpoints for comparison, which is always slightly shorter than the true arc length. For y = x² from x = 0 to x = 1 the arc length is about 1.478943, noticeably longer than the chord of √2 ≈ 1.414214.