TITLE: Differential and Integral calculus.
QUESTION [0 upvotes]: Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and $dh$. When we try to derive the formula for the surface area of a cone, we can take the slant height into consideration but this is not the case when we try to find its volume. Don't you think that the amount of approximation in both the cases is same. I discussed this a lot with my teachers but the discussion was completely against my opinion, so please help. What I think is that the volume of a cone must be somewhere between $\frac{1}{3}\pi r^2h$ and $\frac{1}{3}\pi r^2l$

REPLY [1 votes]: You have to understand that the mass of hollow cone is only at the surface(we consider it's surface to be of thickness $t$ where $t\to0$ because we take product of two infinitesimals $dxdy\to0$) so we can't take vertical height and neglect the surface mass(when taking vertical height neglection is of a thin cone whose thickness  $t\to0$) which we do in solid cone where even if we remove a thin hollow cone, the center of mass remains same( note that the the thickness of removed shell $t\to0$).
Crude Explanation:
When you take vertical height the mass on surface is neglected but when you take vertical height in solid cone of same dimensions, the neglection is very small in comparison to rest of the cone.
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