Let y = f(x) be the graph of a function f, a continuous function on [a,b], and such that f(x) ≥ 0 for all x in [a,b]. The volume V of the solid of revolution obtained by rotating, through one revolution about the x-axis, the region bounded by the curve y = f(x), the x-axis and the lines x = a and x = b, is given by

Parametric form

For the curve x = x(t), y = y(t) (t ε‎ [α‎, β‎]), the volume V is given by

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

Compare area of a surface of revolution