For a curve defined by $y \= f\left(x\right)\>0$ on the interval $a\le x\le b$, the formula for the surface area is given by

$S_x\=2\mathbf{}\mathrm{\pi }\mathbf{}\left({\int }_a^{b}f\left(x\right)\mathbf{}\sqrt{1\+{\left[f \prime \left(x\right)\right]}^2}\ⅆx\right) \=$$2\mathbf{}\mathrm{\pi }\mathbf{}\left({\int }_a^{b}y\mathbf{}\sqrt{1\+{\left(\frac{\ⅆ y}{\ⅆ x}\right)}^2}\ⅆx\right)$

For a curve defined by $x \= g\left(y\right) \> 0$ on the interval $c\le y\le d$, the formula for the surface area is given by

$S_y\=2\mathbf{}\mathrm{\pi }\mathbf{}\left({\int }_c^{d}g\left(y\right)\mathbf{}\sqrt{1\+{\left[g\prime \left(y\right)\right]}^2}\ⅆy\right) \=$$2\mathbf{}\mathrm{\pi }\mathbf{}\left({\int }_c^{d}x\mathbf{}\sqrt{1\+{\left(\frac{\ⅆ x}{\ⅆ y} \right)}^2}\ⅆy\right)$

For a curve defined parametrically by $x\left(t\right)$ and $y\left(t\right)$: