MHB What is the process for finding the centroid of a sliced solid cylinder?

[karush]

Find the centroid.
Sliced Solid Cylinder
bounded by $x^2+y^2=196$,$z=0$,$y+z=14$
so $r=14$
and $r\sin\theta +z=14$
so $z=14-\sin\theta$
$\displaystyle m=\iiint_\limits{D}{}^{} Rv = \int_{0}^{24} \int_{0}^{14} \int_{0}^{14-r\sin\theta}$
$\displaystyle=\int_{0}^{24}\int_{0}^{14}(14r-r^2\sin\theta) \,dr \,d\theta$
$\displaystyle=\int_{0}^{24} (1372-392\sin\theta)d\theta= 2744\pi$

?

the answer is $\displaystyle=\left[0,-\frac{7}{2},\frac{35}{4}\right]$

 

[HallsofIvy]

You have found the volume of the cylinder. You do not appear to have made any attempt to find the centroid. What exactly is your question? Do you know what "centroid" means?

The volume of object "O" is $V= \int\int_O\int dxdydz$. The x coordinate of the centroid is given by $\frac{\int\int_0\int x dxdydz}{V}$, the y coordinate of the centroid is given by $\frac{\int\int_0\int y dxdydz}{V}$ and the z coordinate of the centroid is given by $\frac{\int\int_0\int z dxdydz}{V}$.