Arc Length Calculator (Calculus)

Arc length in calculus is the total length of a curve between two points, computed using integration. For an explicit function y = f(x), the formula is L = ∫[a to b] √(1 + (f′(x))²) dx, where f′(x) is the derivative of the function. This integral sums up infinitely many tiny straight-line segments along the curve.

This calculator uses Simpson's Rule (composite numerical integration) to approximate the definite integral of √(1 + (f′(x))²) over [a, b]. It also evaluates the derivative f′(x) numerically using a central difference formula: f′(x) ≈ (f(x+h) − f(x−h)) / (2h). Increasing the number of intervals improves accuracy.

You can enter most standard mathematical functions, including polynomials (x^2, x^3), square roots (sqrt(x)), trigonometric functions (sin(x), cos(x), tan(x)), exponentials (exp(x) or e^x), and logarithms (log(x) for base-10, ln(x) for natural log). Use ^ for exponentiation and * for multiplication.

For a general curve defined by y = f(x), you don't need a radius — you use the arc length integral formula L = ∫√(1 + (f′(x))²) dx evaluated from a to b. The radius concept only applies specifically to circular arcs, where L = r × θ (with θ in radians).

For a circular arc, arc length is L = r × θ, where r is the radius and θ is the central angle in radians. If your angle is in degrees, convert first: θ (radians) = θ (degrees) × π / 180. For a general calculus curve, radians are used internally when trig functions appear in f(x) or its derivative.

By the triangle inequality, the shortest path between two points is always a straight line. Any curve that deviates from a straight line must travel a longer total distance. The arc length equals the straight-line distance only when f(x) is a linear function (a straight line itself) over the interval.

For a parametric curve x = x(t), y = y(t), the arc length from t = a to t = b is L = ∫[a to b] √((dx/dt)² + (dy/dt)²) dt. Both derivatives are computed with respect to the parameter t, and the integrand represents the speed along the curve at each point.

Accuracy depends on the number of integration intervals selected. Using 1000 intervals typically gives results accurate to 4–6 decimal places for smooth functions. For functions with rapid oscillations or singularities near the interval endpoints, using 5000 intervals and keeping the interval away from undefined points is recommended.