Arc Length of Curve

The hypotenuse is given by $\sqrt{d{y}^2+d{x}^2}=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx$ (the Derivative)
If we do this across the entire interval with smaller and smaller $dx$ and sum them together, we get $\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^n\sqrt{1+{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2}dx$, which is a Riemann Sum, which can be turned into an Integral
$L={\int }_a^b\sqrt{1+{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2}dx$

This is how you might derive the surface area of a Sphere or cone or smth

Surface Area of Revolution

When rotating about some axis

Instead of summing up just the hypotenuse, sum up the circumference of the circle with radius $r=f(x)$ (if rotating about y-axis, use $r=x$):
$SA=2\pi {\int }_a^{b}f(x)\sqrt{1+{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2}dx$