Finding the center of mass of a piecewise function

[JorkThePork]

Homework Statement

Find the center of mass of the "lightning bolt" shaped piecewise function as shown below (or in the desmos project: https://www.desmos.com/calculator/g6crwsecp1)

Relevant Equations

Xcm = A^-1 * ∫ a b x * (f(x) - g(x)) d x

Ycm = A^-1 * ∫ a b .5((f(x)^2 - g(x)^2) d x

I understand that I can divide this shape into a few parallelograms and a triangle and calculate the center of mass of each, but am confused as to what I should do after that. My physics teacher also wants us to use integrals, but I'm assuming I can calculate the COM of each parallelogram and triangle by simply integrating.

 

[kuruman]

You can find the CM of the two parallelograms and the half-parallelogram which will give you 3 point masses at the locations of these CMs. Then you can find the CM of the 3 point masses the usual way.

 

[JorkThePork]

You can find the CM of the two parallelograms and the half-parallelogram which will give you 3 point masses at the locations of these CMs. Then you can find the CM of the 3 point masses the usual way.

the two seemingly big parallelograms actually are not parallelograms because of how I designed the lightning bolt (the equations I used are here https://www.desmos.com/calculator/g6crwsecp1). However, I could break the shape up into 2 big parallelograms, 2 very thin parallelograms, and a triangle. I'm assuming I could still then find the CM of the 5 point masses the usual way.

 

[haruspex]

the two seemingly big parallelograms actually are not parallelograms because of how I designed the lightning bolt (the equations I used are here https://www.desmos.com/calculator/g6crwsecp1). However, I could break the shape up into 2 big parallelograms, 2 very thin parallelograms, and a triangle. I'm assuming I could still then find the CM of the 5 point masses the usual way.

Yes, you can carve it up into simple non overlapping shapes, but since you have equations why not just integrate them and add the (signed) results?

 

[JorkThePork]

Yes, you can carve it up into simple non overlapping shapes, but since you have equations why not just integrate them and add the (signed) results?

Sorry, I’m a little confused how this would give me the center of mass. wouldn’t this just give the area?

 

[haruspex]

Sorry, I’m a little confused how this would give me the center of mass. wouldn’t this just give the area?

So what do you need to multiply the expressions by before integrating?