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In a lot of math formulas, I usually try to derive the formula by trying to reach rough approximation of the equation before looking at the actual equation itself (nearly 99% of the time I never get the equation but I get an idea).

For the surface of revolution, I had my go at deriving the equation but I'm still wondering where is my downfall in my logic.

Approach: If we have a function revolve around, say the x axis, we get a surface. We can divide this surface into small circular bands of area (perpendicular to axis) approximately equal to 2πrΔL (Lateral area of cylinder). Take the summation to find the integral.

What is wrong with my approach?

Ian L
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  • I just realized what might be my error: ΔL is not dx. So I have to rewrite ΔL as a function of x. Following a similar approach to the length of a curve, you will obtain the answer. Is this correct? – Ian L Jun 07 '16 at 04:09
  • Yes, if for example the curve makes a $45$ degree angle with the axis of revolution, then the little surface area is about $2\pi \sqrt{2},dx$. As you point out, the same issue arises with arclength. – André Nicolas Jun 07 '16 at 04:31

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