How to to choose approximation for calculus/integral calculation? (example for surface of revolution) To calculate the surface of a revolution (spinning around the x axis), we usually use the surface of a circular truncated cone (labeled $A$) to approximate the infinitesimal of the surface of the revolution.

The result is all known
$$
S=2\pi \int_a^b f(x) \sqrt{1+{f'}^2(x)}dx
$$
But if we choose a cylinder to approximate the infinitesimal, the result will be a wrong equation
$$
S=2\pi \int_a^b f(x) dx
$$

So my question is, is there a rigid principal to choose the approximation? How can we be sure that the approximation is correct, how to validate?

We have the same problem in the simpler situation of a graph $$\gamma:\quad y=f(x)>0\qquad(a\leq x\leq b)\ .$$
When we want to approximate the area under this graph we can take rectangles cut off horizontally at the top, but for the length of $\gamma$ we have to stay with hypotenuses, which then lead to the nasty square root under the integral sign.
The reason can be seen in the following figure: When we split a rectangle vertically in half the relative area-error committed by neglecting the triangles is halved as well, and therefore goes to zero in the limit. But with the hypotenuses it's different: There is always a factor $${1\over\cos\theta}=\sqrt{1+\tan^2\theta}$$ between the width of the rectangle at stake and the length of the corresponding hypotenuse. This factor does not go away in the limit.

