Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. However, if we want to find the surface area, then we approximate it as part of a cone and the formula is $$\int_a^b 2\pi f(x)\sqrt{1+f'(x)^2} dx.$$ But if approximated it by a circle with thickness $dx$ we would get $$\int_a^b 2\pi f(x) dx.$$
So my question is how come for volume we can make the cruder approximation of a disk but for surface area we can't.