MHB Where is the Center of Mass of a Thin Plate with Given Boundaries?

[karush]

$\textsf{Find the center of mass of a thin plate of density}$
$\textsf{ $\delta=3$ bounded by the lines $x=0, y=x$, and the parabola
$y=2-x^2$ in the $Q1$}$
$\begin{array}{llcr}\displaystyle
&\textit{Mass}\\
&&\displaystyle M=\iint\limits_{R}\delta \, dA\\
&\textit{First Moments}\\
&&\displaystyle M_y=\iint\limits_{R}x\delta \, dA
&\displaystyle M_x=\iint\limits_{R}y\delta \, dA\\
&\textit{Center of mass}\\
&&\displaystyle\bar{x}=\displaystyle\frac{M_y}{M},
\displaystyle\bar{y}=\displaystyle\frac{M_x}{M}\\
\\
&&\color{red}
{\displaystyle \, \bar{x}=\frac{5}{14},
\bar{y}=\displaystyle\frac{38}{35}}\\
\end{array}$ok I just barely had to time to post this
equations are just from reference
red is answer

 

[karush]

Ok I'm starting over on this a small step at a time...

$\textsf{Find the center of mass of a thin plate of density}\\$
$\textsf{$\delta=3$ bounded by the lines $x=0, y=x$, and the parabola
$y=2-x^2$ in $Q1$}\\$
\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}\\
\end{align*}

something isn't happening right!


$\textsf{the answer utimately is:}\\$
$\color{red}{\, \bar{x}=\displaystyle\frac{5}{14},\bar{y}=\frac{38}{35}}$

 

[MarkFL]

The first thing I would do is sketch the bounded area:

View attachment 7772

Now, let's compute the mass (noting that the curves $y=x$ and $y=2-x^2$ intersect at $x=1$ in QI):

$$m=\rho A=3\int_{0}^{1}\int_{x}^{2-x^2}\,dy\,dx=3\int_{0}^{1}2-x-x^2\,dx=3\left(2-\frac{1}{2}-\frac{1}{3}\right)=\frac{7}{2}$$

Next, let's compute the moments of the lamina:

$$M_x=3\int_{0}^{1}\int_{x}^{2-x^2}y\,dy\,dx=\frac{3}{2}\int_{0}^{1}\left(2-x^2\right)^2-x^2\,dx=\frac{3}{2}\int_{0}^{1}x^4-5x^2+4\,dx=\frac{19}{5}$$

$$M_y=3\int_{0}^{1}x\int_{x}^{2-x^2}\,dy\,dx=3\int_{0}^{1}2x-x^2-x^3\,dx=\frac{5}{4}$$

Hence:

$$\overline{x}=\frac{M_y}{m}=\frac{\frac{5}{4}}{\frac{7}{2}}=\frac{5}{14}$$

$$\overline{y}=\frac{M_x}{m}=\frac{\frac{19}{5}}{\frac{7}{2}}=\frac{38}{35}$$

 

[karush]

View attachment 7773

ok i graphed this but
it not it

why

 

[skeeter]

$\textsf{Find the center of mass of a thin plate of density}$
$\textsf{ $\delta=3$ bounded by the lines $\color{red}{x=0}$, $ y=x$, and the parabola
$y=2-x^2$ in the $Q1$}$

$\color{red}{x=0}$ is the y-axis as shown in Mark’s sketch, not the x-axis as shown in your sketch.