Solutions - 0110-03

Assume $\rho =1$ for all of these.

The value of $\rho$ does not affect the CoM point if $\rho$ is a constant.

Region 1:

$$\begin{array}{c}{m}_1=\frac{1}{2}(2)(1)\gg \gg 1\\ \\ M_{1y}=0\\ \\ M_{1x}={\int }_8^{9}y(2+\frac{0-2}{1}y)dy\gg \gg \frac{25}{3}\end{array}$$

Region 2:

$$\begin{array}{c}{m}_2=\pi {(1)}^2\gg \gg \pi \\ \\ M_{2y}=0\\ \\ M_{2x}=m_2{\bar{y}}_2\gg \gg \pi (7)\gg \gg 7\pi \end{array}$$

Region 3:

$$\begin{array}{c}{m}_3={\int }_{-2}^2(6-(2+x^2))dx\gg \gg \frac{32}{3}\\ \\ M_{3y}=0\\ \\ M_{3x}={\int }_{-2}^2\frac{1}{2}(6^2-{(2+x^2)}^2)dx\gg \gg \frac{704}{15}\end{array}$$

Region 4:

$$\begin{array}{c}{m}_4=2\left(\frac{1}{2}\right)\gg \gg 1\\ \\ M_{4y}=m_4{\bar{x}}_4\gg \gg -1\left(\frac{9}{4}\right)\gg \gg -\frac{9}{4}\\ \\ M_{4x}=m_4{\bar{y}}_4\gg \gg 1(5)\gg \gg 5\end{array}$$

Region 5:

$$\begin{array}{c}{m}_5=2\left(\frac{1}{2}\right)\gg \gg 1\\ \\ M_{5y}=m_5{\bar{x}}_5\gg \gg 1\left(\frac{9}{4}\right)\gg \gg \frac{9}{4}\\ \\ M_{5x}=m_5{\bar{y}}_5\gg \gg 1(5)\gg \gg 5\end{array}$$

Region 6:

$$\begin{array}{c}{m}_6=3\left(\frac{1}{2}\right)\gg \gg \frac{3}{2}\\ \\ M_{6y}=m_6{\bar{x}}_6\gg \gg -\frac{3}{2}\left(\frac{5}{4}\right)\gg \gg -\frac{15}{8}\\ \\ M_{6x}=m_6{\bar{y}}_6\gg \gg \frac{3}{2}\left(\frac{3}{2}\right)\gg \gg \frac{9}{4}\end{array}$$

Region 7:

$$\begin{array}{c}{m}_7=3\left(\frac{1}{2}\right)\gg \gg \frac{3}{2}\\ \\ M_{7y}=m_7{\bar{x}}_7\gg \gg +\frac{3}{2}\left(\frac{5}{4}\right)\gg \gg \frac{15}{8}\\ \\ M_{7x}=m_7{\bar{y}}_7\gg \gg \frac{3}{2}\left(\frac{3}{2}\right)\gg \gg \frac{9}{4}\end{array}$$

FlatCoMMan:

$$\begin{array}{c}m=\frac{50}{3}+\pi ,M_x=\frac{2093+210\pi }{30},M_y=0\\ \\ \gg \gg (\bar{x},\bar{y})=(0,\frac{2093+210\pi }{500+30\pi })\approx 4.63\end{array}$$