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I'm a lowly Calculus II student here and I noticed something interesting/confusing.

The integral formula for the volume of a solid of revolution (SoR) performed with the disk method has the form $$ V=\int\pi f(x)^2 \mathrm{d}x, $$ and the same holds for the formula for the area of a circle, but this time $f(x)$ is the radius. This makes a lot of intuitive sense.

From this, it would also make sense that the formula for the surface area of a SoR is the integral of the length of the circumference of a circle but with $f(x)$ as the radius again. However, this is not the case. The expression for surface area of a SoR is the circumference arc length.

Why is this the case?

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Imagine how the cone is a special surface of revolution and what we can do to generalize.

Surface area of a cone of slant generator length $l$ maximum radius $r$ and constant cone angle $\sin^{-1}\dfrac{r}{l}$ is

$$=\pi r l$$

When cone angle changes ( $r$ changes with $l$) you should have curved slant area $$\int 2 \pi r dl$$

Here you are summing up changing surface area of cones instead of volumes.

Factor 2 comes in to take care of varying radius along instantaneous slanted generators.

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