Find the volume of the solid formed by rotating the region enclosed by
$$x=0,x=1,y=0,y=9+x^2$$
about the $y$-axis.

Step 1 ：First, we need to understand that the solid is formed by rotating the region enclosed by the given equations about the y-axis. This means that the solid is a series of cylindrical shells, each with a radius equal to the x-coordinate, a height equal to the difference between the upper and lower y-values, and a thickness of dx.

Step 2 ：The volume of each cylindrical shell is given by the formula $2\pi xhdx$, where x is the radius, h is the height, and dx is the thickness.

Step 3 ：In this case, the radius x ranges from 0 to 1, the height h is given by the equation $9+x^2$, and the thickness dx is a small change in x.

Step 4 ：We can find the total volume of the solid by integrating the volume of each cylindrical shell from x=0 to x=1. This gives us the integral ${\int }_0^{1}2\pi x(9+x^2)dx$.

Step 5 ：We can simplify this integral by distributing the $2\pi x$ to get ${\int }_0^{1}18\pi x+2\pi x^{3}dx$.

Step 6 ：We can then evaluate this integral by finding the antiderivative of the integrand and evaluating it at the limits of integration. The antiderivative of $18\pi x$ is $9\pi x^2$, and the antiderivative of $2\pi x^3$ is $\frac{1}{2}\pi x^4$.

Step 7 ：Evaluating these at the limits of integration gives $(9\pi (1{)}^2+\frac{1}{2}\pi (1{)}^4)-(9\pi (0{)}^2+\frac{1}{2}\pi (0{)}^4)=9\pi +\frac{1}{2}\pi$.

Step 8 ：Adding these together gives the total volume of the solid as $\boxed{{\displaystyle \frac{19}{2}\pi }}$ cubic units.