A rod on $[0,4]$ has linear density $\rho (x)=3x$. Find the exact center of mass $\bar{x}$.

A rod on $[1,5]$ has linear density $\rho (x)=2x+1$. Find the exact center of mass $\bar{x}$.

A rod on $[0,2]$ has linear density $\rho (x)=4-x$. Find the exact center of mass $\bar{x}$.

A rod on $[-1,2]$ has linear density $\rho (x)=x+3$. Find the exact center of mass $\bar{x}$.

A rod on $[0,3]$ has linear density $\rho (x)=x^2+1$. Find the exact center of mass $\bar{x}$.

For the uniform lamina under $y=\sqrt{x}$ on $[0,9]$, which integral gives the moment $M_x$ about the $x$-axis?

For the uniform lamina under $y=2x+1$ on $[0,3]$, which integral gives the moment $M_y$ about the $y$-axis?

For the uniform lamina under $y=4-x^2$ on $[-2,2]$, which integral gives the moment $M_x$ about the $x$-axis?

For the uniform lamina under $y=1+x^2$ on $[0,1]$, which integral gives the moment $M_y$ about the $y$-axis?

For the uniform lamina under $y=\sin x$ on $[0,\pi ]$, which integral gives the moment $M_x$ about the $x$-axis?

Find the exact $\bar{x}$ for the uniform lamina bounded by $y=x$ and $y=x^3$ on $[0,1]$.

Find the exact $\bar{x}$ for the uniform lamina bounded by $y=4-x^2$ and the $x$-axis on $[0,2]$.

Find the exact $\bar{x}$ for the uniform lamina bounded by $y=2$ and $y=x$ on $[0,2]$.

Find the exact $\bar{x}$ for the uniform lamina bounded by $y=x+1$ and $y=x^2+1$ on $[0,1]$.

Find the exact $\bar{y}$ for the uniform lamina bounded by $y=3x$ and $y=x^2$ on $[0,3]$.

Find the exact $\bar{y}$ for the uniform lamina under $y=\sqrt{x}$ on $[0,4]$.

Find the exact $\bar{y}$ for the uniform lamina bounded by $y=2$ and $y=\frac{x^2}{2}$ on $[-2,2]$.

A uniform lamina is bounded by $y=9-x^2$ and the $x$-axis on $[-3,3]$. What is $\bar{x}$, and why?

A uniform lamina is bounded by $x=y^2$ and $x=4$ for $-2\le y\le 2$. What is $\bar{y}$, and why?

A uniform lamina is bounded by $x=(y-1{)}^2$ and $x=4$ for $-1\le y\le 3$. What is $\bar{y}$, and why?