Difference in centroid, disc and shells There are formulas for calculating the centroid of a shape.
$$Centroid/Strips: (x,y)=(\frac{\int xy dx}{\int y dx} , \frac{0.5\int y^2 dx}{\int y dx} )$$
$$Discs: (x,y)=(\frac{\int xy^2 dx}{\int y^2 dx} , 0 )$$
$$Shell: (x,y)=( \frac{\int x\sqrt{1+(\frac{dy}{dx})^2}dx}{\int \sqrt{1+(\frac{dy}{dx})^2}dx}, \frac{\int y\sqrt{1+(\frac{dy}{dx})^2}dx}{\int \sqrt{1+(\frac{dy}{dx})^2}dx}
)$$
I tried to search on Google about what they're implying but each page is about the derivation of formula of Centroid using Equilibrium of moments.
Can somebody please explain when are these formulas supposed to be used?(Maybe explain the distinction between Centroid/Strips, Disc and shells)
 A: This kind of formulaic teaching is disturbing. If you understand the definition of the centroid then you can attack any such problem. To that end we define the centroid as
$$\mathbf{R}=\frac{\int_D\mathbf{r}dD}{\int_DdD}$$
where $D$ is the domain in question; it can be a line, area, or volume and $\mathbf{r}=(x,y,z)$, as appropriate. This is seen to be a weighted average of the incremental elements with respect ot the origin.
So let's get specific, for a line, the domain is $S$ and
$$\mathbf{R}=\left(\frac{\int xds}{S},\frac{\int yds}{S} \right)$$
where $S$ is the line length and $ds=\sqrt{1+\left(\frac{dy}{dx}^2 \right)}\,dx.$
For an area, the domain is the area $A$ and
$$\mathbf{R}=\left(\frac{\int\int xdydx}{A},\frac{\int\int ydydx}{A} \right)$$
And finally, for a volume, $V$
$$\mathbf{R}=\left(\frac{\int\int\int xdzdydx}{V},\frac{\int\int\int ydzydx}{V},\frac{\int\int\int zdzdydx}{V} \right)$$
And the last thing you need to know is that when when you have compound shapes, for example, a square with circular cap, or a disk with hole in it, the centroid is the weighted averaged of the centroids of the components. For example,
$$R=\frac{R_1A_1+R_2A_2+R_3A_3+\cdots}{A}$$
where $A=\sum A_k$. If any of the areas are holes then they are simply negative areas.
With these definitions you should be able to calculate the centroid of any shape.
