Why do we need the function $y= f(x)$ to be non-negative to calculate surface area of revolution? Can't we just substitute $|f(x)|$ in the formula?

Question

I was looking at the derivation of the surface area of the revolution of a graph. All the sources I refer to use non-negative functions, and use dividing the revolution surface into frustums to get the result. I don't see how the derivation does not hold for if the function is negative at some point. we just have to get a finer partition in the derivation such that the point at which the graph goes to zero is a partition point and the rest of the derivation remains the same. Am I doing something wrong? Specifically, is there something wrong with using $$\int^b_a2\pi|f(x)| \sqrt{1+f'(x)^2}$$ instead of $$\int^b_a2\pi f(x) \sqrt{1+f'(x)^2}$$.