Finding the center of mass of a lamina using polar coordinates

[burt]

I have been working on the following question:

Here is my work:




To me, I obviously did something wrong here. I don't think it makes sense for my center of mass to be at (0,0). So, I'm trying to figure out my mistake. My first place to look is at my limits of integration. In order to figure out the limits for r, I found the biggest r and the smallest r by putting 0 and \(\pi\) in for \(\theta\). Should I have used the equation instead as a limit?

Also, was my technique correct? This is the technique from the Cartesian system - perhaps there is something different I am supposed to do when using polar equations?

And, of course, it could be there is a different problem that I haven't yet found.

Any guidance is appreciated!
Thank you!

 

[burt]

I have been reviewing this problem trying to find my mistake. I changed my limits of integration and got \((-\frac{12}{5},0)\) as my center of mass - which intuitively makes more sense - I'll attach a picture of the graph. Here is my corrected work:


Is this correct? What was wrong with my original method?

Thank you!

(the graph is attached as an attachement)