Let y = f(x) be the graph of a function f such that f′ is continuous on [a, b]. The length of the arc, or arc length, of the curve y = f(x) between x = a and x = b equals

Parametric form

For the curve x = x(t), y = y(t) (t ∈ [α‎, β‎]), the arc length equals

$${\displaystyle {\int }_{\alpha }^{\beta }\sqrt{{\left(\frac{\text{d}x}{\text{d}t}\right)}^2+{\left(\frac{\text{d}y}{\text{d}t}\right)}^2}\text{d}t.}$$

Polar form

For the curve r = r(θ‎) (α‎ ≤ θ‎ ≤ β‎), the arc length equals

$${\displaystyle {\int }_{\alpha }^{\beta }\sqrt{r^2+{\left(\frac{\text{d}r}{\text{d}\theta }\right)}^2}}\text{d}\theta .$$