Let y = f(x) be the graph of a function f such that f′ is continuous on [a,b] and f(x) ≥ 0 for all x in [a,b]. The area of the surface obtained by rotating, through one revolution about the x-axis, the arc 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 surface area equals

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

Polar form

For the curve r = r(θ‎) (α‎ ≤ θ‎ ≤ β‎), the surface area equals

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