Arc Length Integral Calculator
The arc length of a smooth curve y = f(x) is the integral of the square root of 1 plus the slope squared. Enter the bounds, an even number of subintervals, and the derivative dy/dx at each evenly spaced node. The calculator builds the arc length integrand and integrates it with Simpson's rule to return the curve length.
Arc length formula
L = integral from a to b of sqrt(1 + (dy/dx)^2) dx
integrand at node i: g_i = sqrt(1 + (dydx_i)^2)
h = (b - a) / n
L = (h / 3) * [ g0 + gn + 4*(odd g) + 2*(even g) ] (Simpson)
The integrand is at least 1 everywhere, so the arc length is never less than the straight horizontal distance b minus a. Supply the derivative at n plus 1 evenly spaced nodes.
Worked example
For y = x squared from 0 to 1, dy/dx = 2x, giving derivative values 0, 0.5, 1, 1.5, 2 at the five nodes with h = 0.25. The integrand values are 1, 1.1180, 1.4142, 1.8028, 2.2361. Simpson's rule gives an arc length of about 1.4789, close to the exact value of roughly 1.4789.
Arc length: frequently asked questions
What is the arc length formula?
For a curve y = f(x) from x = a to x = b, the arc length is the integral from a to b of the square root of 1 plus (dy/dx) squared, with respect to x. It sums the lengths of infinitesimal slope-adjusted segments along the curve.
How does this calculator evaluate it?
You supply the derivative dy/dx at evenly spaced nodes from a to b. The calculator forms the integrand, the square root of 1 plus the derivative squared, at each node, then integrates with composite Simpson's rule for an accurate numeric arc length.
How many derivative values do I enter?
Enter n plus 1 values for an even number of subintervals n, one derivative at each node from the lower bound to the upper bound inclusive. The nodes must be evenly spaced, matching the step size the calculator computes from the bounds.
Why is the integrand never less than one?
Because it is the square root of 1 plus a squared quantity, the integrand is at least 1. The arc length is therefore always at least the straight horizontal distance b minus a, with equality only for a flat line.
Sources
- NIST Digital Library of Mathematical Functions: Calculus of one variable, arc length.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.