$L=\int_{\alpha}^{\beta}\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$

The formula to find the surface area is very similar.

If the curve is rotating around the $x$-axis, where $f', g'$ are continuous and $g(t) \geq 0$, then the formula for the surface area of the curve is

$SA=\int_{\alpha}^{\beta} 2\pi y\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$