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I will explain my question with an example:

Let's take two surfaces that have the same area:

SURFACE A $$ f(x) = x, x \in [0,6] $$ enter image description here

SURFACE B $$ f(x) = 3, x \in [0,6] $$ enter image description here

Both surfaces are equal:

Surface A, Sa = 18
Surface B, Sb = 18

But now if we rotate those two surfaces around the (for example) x axis.

enter image description here

We obtain two different volumes:

Volume A, Va = 72*pi
Volume B, Vb = 54*pi

I've no problem to apply the formula, but I found the result a bit counter-intuitive. We apply the "same" area (with different shape) around an axis an we obtain a different volume.

Why is that (intuitively) ?

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  • $\begingroup$ If you rotate the triangle about the $y$-axis, you don't get a cone. $\endgroup$
    – Allawonder
    Aug 26 '19 at 16:03
  • $\begingroup$ @Allawonder, thank you, I've edited my question $\endgroup$
    – obchardon
    Aug 26 '19 at 16:11
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Even though they have the same area, their centers of mass are very different, which should be the base for comparison. The area with the center of mass further away from the y-axis will travel around the y-axis in a bigger circle, hence collecting more volume.

Since the triangle has its center further than that of the rectangle, it has bigger volume from the rotation. In fact, the resulting volume is proportional to the circumference of the circle traveled by the center. For the example given, the center of mass is 4 units from the y-axis for the triangle and 3 units for the rectangle. Therefore, their ratio of the rotating volumes is 4:3.

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There is a result that says that equal areas rotated about axes an equal distance away from the center-of-mass (or centroid) of the areas, produce equal volumes. If the area is not distributed equally with respect to the distance from the axis of rotation, you will get different volumes.

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