Find center of mass given Lamina pictured: https://s3.amazonaws.com/wamapdata/qimages/qtrring.gif
with inner radius of 3 and an outer radius of 7, and a density function $$\rho(x,y) = \frac{x}{x^2+y^2}.$$
My Assumptions: $$0 < \theta < \frac{\pi}{2}, \quad 3 < r < 7.$$
Originally I went with polar coordinates and got $M = 4, M_x = 2, M_y = \pi$. It was incorrect. Can anyone tell me what the answer is and what I did wrong?
We note that we have generically for the mass of a plane object $\Sigma$ we have:
$$M=\iint_{\Sigma}\rho(\vec{r})\:\mathrm{d}A$$
In this case we have that: $x=r\cos(\theta),y=r\sin(\theta)$,where $r\in[3,7]$ and $\theta\in[0,\frac{\pi}{2}]$. And thus:
$$M=\int_{3}^{7}\int_{0}^{\frac{\pi}{2}}\frac{r\cos(\theta)}{r^{2}}\:r\:\mathrm{d}\theta\:\mathrm{d}r=\int_{3}^{7}\:\mathrm{d}r=7-3=4$$
We also note that:
$$\begin{pmatrix}\overline{x} \\ \overline{y}\end{pmatrix}=\frac{1}{M}\iint_{\Sigma}\vec{r}\rho(\vec{r})\:\mathrm{d}A$$
Using the same bounds as before we get:
$$\begin{pmatrix}\overline{x}\\ \overline{y}\end{pmatrix}=\frac{1}{4}\int_{3}^{7}\int_{0}^{\frac{\pi}{2}}\begin{pmatrix}r\cos(\theta)\cos(\theta) \\ r \sin(\theta)\cos(\theta)\end{pmatrix}\:\mathrm{d}\theta\:\mathrm{d}r=\frac{1}{4}\int_{3}^{7}\begin{pmatrix}\frac{\pi r}{4} \\ \frac{r}{2}\end{pmatrix}\:\mathrm{d}r=\begin{pmatrix}\frac{5\pi}{4} \\ \frac{5}{2}\end{pmatrix}$$