Mnemonic for centroid of a bounded region

Question

The centroid of a region bounded by two curves is given by:

$ \bar{x} = \frac{1}{A}\int_a^b{x\left[f(x)-g(x)\right]dx} $

$ \bar{y} = \frac{1}{A}\int_a^b{\left[\frac{(f(x)+g(x)}{2}(f(x)-g(x))\right]dx} = \frac{1}{2A}\int_a^b{\left(f^2(x) - g^2(x)\right)dx}$

where A is just the area of that region.

But I have a terrible time remembering those formulas (when taken in conjunction with all of the other things that need to be remembered), and which which moment uses which formula. Does anybody know a good mnemonic to keep track of them?

Hopefully this isn't off topic. Thanks

Answer 1

In order to remember those formulas, you have to use them repeatedly on many problems involving finding centroid of areas bounded by two curves. Have faith in the learning process and you will remember it after using it many times, just like playing online games.

Answer 2

The mnemonics are $$\overline x=\frac1A\int\int_Rx\,dy\,dx$$ and $$\overline y=\frac1A\int\int_Ry\,dy\,dx$$ where $R$ is the region and $A=\int\int_R1\,dy\,dx.$