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Before I read the formula of the area of revolution which is $\int 2\pi y \,ds$, where $ds = \sqrt{1 + \frac{dy}{dx}^2}$, I thought of deriving it myself. I tried to apply the same logic used for calculating the volume of revolution (e.g., $\int \pi y^2 dx $).

My idea is to use many tiny hollow cylinders (inspired from the shell method), each has a surface area of $(2\pi y) (dx)$:

  • $2\pi y$ is the circumference of the cylinder, and
  • $dx$ is the height of the cylinder

Their product is the surface area of the hollow (e.g., empty from the inside) cylinder.

With this logic, the area is $\int 2\pi y dx$.

Where is my mistake? Also it's confusing why for the volume it was enough to partition the object using cylinders and for areas not.

user2232305
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1 Answers1

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The problem is that you can’t approximate at the first order the surface with a cylinder. A cylinder has tangent planes orthogonal to the $xy$ plane. This is not the case for a general surface whose tangent plane may be oblique.

You need to approximate the surface with a circular cone. In which case, you come back to the given formula.

mathcounterexamples.net
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  • What disturbs me is that using cylinders work for calculating the volume. Why aren't we using cones to calculate the volume as well? The intuition is that dx is so small that what is between x and x+dx is parallel to the x plane. – user2232305 Apr 08 '21 at 05:13
  • The answer in this post answers my question. – user2232305 Apr 08 '21 at 05:37