Use the shell method to find the volume generated by revolving the shaded region about the $y$-axis.
Set up the integral that gives the volume of the solid.

Step 1 ：The shaded region can be divided into two parts: a rectangle with a height of 5 units and a width of 1 unit, and a rectangle with a height of 2 units and a width of 3 units.

Step 2 ：The volume of the solid generated by revolving the first rectangle about the $y$-axis can be calculated using the shell method. The formula for the volume of a cylindrical shell is $2\pi rh$, where $r$ is the distance from the axis of rotation to the center of the shell, and $h$ is the height of the shell. In this case, $r$ varies from 0 to 1, and $h$ is always 5. So the volume is ${\int }_0^{1}2\pi rhdr={\int }_0^{1}2\pi r(5)dr=10\pi {\int }_0^{1}rdr=10\pi [\frac{1}{2}{r}^2{]}_0^1=5\pi$ cubic units.

Step 3 ：The volume of the solid generated by revolving the second rectangle about the $y$-axis can be calculated in a similar way. In this case, $r$ varies from 1 to 4, and $h$ is always 2. So the volume is ${\int }_1^{4}2\pi rhdr={\int }_1^{4}2\pi r(2)dr=4\pi {\int }_1^{4}rdr=4\pi [\frac{1}{2}{r}^2{]}_1^4=4\pi (\frac{1}{2}(4^2-1))=12\pi$ cubic units.

Step 4 ：Therefore, the total volume of the solid is $5\pi +12\pi =\boxed{{\displaystyle 17\pi }}$ cubic units.