Centre of mass of a frustum of a cone How do I find the centre of mass of a solid frustum of a cone?
Do I try it in a manner similar to finding the centre of mass of a solid cone? However I'm not sure what limits of integration do I take while trying to solve in that way.
 A: Let $a$ and $b$ be the radii of your conical frustum, and let $h$ be its height. To find the $z$-coordinate of the centre of mass by integration, you need to find radius at height $z$, denoted by $r(z)$.

The usual way is to extend the frustum to the cone of height $H$. Then from similar triangles you find
$$\frac{H}a =\frac{H-h}b=\frac{H-z}{r(z)}$$
from where you can calculate $H$ and then $r(z)$.
A more interesting way for find $r(z)$ is solving differential equations. Notice that the area of the gray trapezium can be expressed in two ways as
$$\int_0^z r(z')\,dz' = z\cdot \frac{a+r(z)}2$$
so taking the derivative $\frac{d}{dz}$ gives
$$r(z)=\frac{a+r(z)}2+\frac12z\frac{dr}{dz} \implies r(z)=z\frac{dr}{dz}+a \implies \frac{dz}{z} = \frac{dr}{r-a}.$$
Now integrating from $z$ to $b$ gives
$$\int_{z'=z}^b \frac{dz'}{z'} = \int_{r'=r(z)}^b \frac{dr'}{r'-a} \implies \ln \frac{h}z = \ln \frac{b-a}{r(z)-a} \implies r(z)=a+(b-a)\frac{z}h.$$
Finally, you can calculate the volume in cylindrical coordinates
$$V = \int dV = \int_{\phi=0}^{2\pi} \int_{z=0}^h\int_{r=0}^{r(z)} r\,dr\,dz\,d\phi = \pi \int_0^h r(z)^2\,dz$$
$$z_{\text{cm}} = \frac1V\int z\,dV = \frac1V\int_{\phi=0}^{2\pi} \int_{z=0}^h\int_{r=0}^{r(z)} zr\,dr\,dz\,d\phi = \frac{\pi}{V} \int_0^h zr(z)^2\,dz.$$
A: How about looking at the frustum as being a smaller cone subtracted from the larger cone. If we already know the location of COM for a solid cone, its just that,
(COM of larger cone)(mass of larger cone) = (COM of smaller cone)(mass of smaller cone) + (COM of frustum)*(mass of frustum)
For uniform density, we can write "volume" instead of "mass" with no loss.
If you want to do it with integration, with the variable of integration being the "height" of the shape, then the limits will be from 0 to b in contrast for a cone being from 0 to h. were, h -> height of full cone  b->height of frustum
