# Thread: Find the center of mass of a thin plate of density

1. ## Find the center of mass of a thin plate of density

$\textsf{Find the center of mass of a thin plate of density$\delta=3$bounded by the lines$x=0, y=x$, and the parabola$y=2-x^2$in the$Q1$}\\$
\begin{align*}\displaystyle
M&=\int_{0}^{\sqrt{2}}\int_{2-x^2}^{1}3 \, dy \, dx\\
&=3\int_{0}^{\sqrt{2}}\biggr[y\biggr]_{y=2-x^2}^{y=1}\, dx
=3\int_{0}^{\sqrt{2}}(x^2-1) \, dx
=3\biggr[\frac{x^3}{3}-x\biggr]_0^{\sqrt{2}}=\sqrt{2}\\
M_y&=\int_{0}^{\sqrt{2}}\int_{2-x^2}^{1} \, dy \, dx
=\int_{0}^{\sqrt{2}}\biggr[y\biggr]_{y=2-x^2}^{y=1}\, dx\\
&=\int_{0}^{\sqrt{2}}(x^2-1) \, dx = -\sqrt{2}\\
&\color{red}{\, \bar{x}=\frac{5}{14},\bar{y}=\frac{38}{35}}
\end{align*}

ok not doing something right

2. ## Re: Find the center of mass of a thin plate of density

try

$\int _0^1\int _x^{2-x^2}$

4. ## Re: Find the center of mass of a thin plate of density

here is eventually what I did