Let $T$ be the region bounded by the graph of $f\left(x\right)=\frac{1}{x+3}$, the $x$-axis, and the vertical lines $x=2$ and $x=6$. Find the volume of the solid formed when $T$ is revolved about the $y$-axis.

Let $f\left(x\right)=\sqrt{2x+3}$. Determine the area of the surface generated when the region bounded by $f\left(x\right)$ and the $x$-axis on the interval $[1,2]$ is revolved about the $x$-axis.

Calculate the volume of the solid formed by revolving the region bounded by $y=\sin \left(x\right){\cos }^{\frac{1}{2}}\left(x\right)$ and the $x$-axis over $[0,\frac{\pi }{2}]$ about the $x$-axis.

Determine the arc length of the curve $x=5y-2$ from $y=-1$ to $y=3$.

Let $R$ be the region bounded by $y=2-x^2,x=0$, and $y=0$ in the first quadrant. Use the shell method to find the volume of the solid generated when $R$ is revolved about the $y$-axis.