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 $\int_{0}^{1} 2\pi rh \, dr = \int_{0}^{1} 2\pi r(5) \, dr = 10\pi \int_{0}^{1} r \, dr = 10\pi [\frac{1}{2}r^2]_{0}^{1} = 5\pi$ 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 $\int_{1}^{4} 2\pi rh \, dr = \int_{1}^{4} 2\pi r(2) \, dr = 4\pi \int_{1}^{4} r \, dr = 4\pi [\frac{1}{2}r^2]_{1}^{4} = 4\pi (\frac{1}{2}(4^2-1)) = 12\pi$ cubic units.

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