I don't quite understand why the formula to find the surface area of a revolution is what it is:
$$A = 2\pi \int_a^b x\ \sqrt{1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2}\ \text{d} x.$$
I would've though that A could equal: $A = 2\pi \int_a^b y\ \text{d} x$.
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ΔS / | Δy
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Δx
I've looked at proofs but I don't really understand why $\Delta s\to 0$ and not $\Delta x\to 0$. I would've thought that as $\Delta x\to 0$ (or $\Delta y\to 0$) you'd end up with a slice infinitelly small, and it wouldn't have a slope to it, then I would've though that you could add up the perimeter of these 'slopeless' slices, and get the surface area.
I'd would help if I could geometrically see what is going wrong with letting $\Delta x\to 0$ (or $\Delta y\to 0$).
More specifically, if I just go $2\pi y$, what am I not taking into account (graphically).
I know this is like the arc length (length of a curve) formula, I have the same grievance with that.
Thank you.