The region of the triangle is bounded by the lines $y = 2x$, $y = 9$, and $x = 0.5$, and we are rotating this region around the y-axis. To calculate the volume of the solid formed by this rotation using the method of cylindrical shells, we will set up the integral with respect to $y$ (since we are using $dy$).

Step-by-step process:

1. Equation of the boundary lines:

   • $y = 2x$ can be rewritten as $x = \frac{y}{2}$.
   • The vertical line is $x = 0.5$.
   • The horizontal line is $y = 9$.

2. Set up the integral using cylindrical shells: The formula for the volume using the shell method when rotating around the $y$-axis is: $V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} (x \cdot \text{height}) \, dy$ where $x$ is the distance from the axis of rotation (the y-axis), and the height of each shell is the corresponding width of the triangle at a given $y$.

   • The height (or the width of the triangle at a given $y$) is $0.5 - \frac{y}{2}$, because at a given $y$, the right boundary of the triangle is at $x = 0.5$ and the left boundary is $x = \frac{y}{2}$.

3. Integral bounds:

   • The lower bound for $y$ is at $y = 0$ (where the triangle starts).
   • The upper bound is at $y = 9$ (where the horizontal line $y = 9$ intersects the region).

4. Set up the integral: The volume $V$ is given by: $V = 2\pi \int_0^9 \left( \left( 0.5 \right) - \frac{y}{2} \right) \, dy$

5. Evaluate the integral: $V = 2\pi \left[ 0.5y - \frac{y^2}{4} \right]_0^9$

   Substitute the bounds into the expression:

   [ V = 2\pi \left( \left[ 0.5(9) - \frac{9^2}{4} \right] - \left[ 0.5(0) - \frac{0^2}{4} \right] \right) ] $V = 2\pi \left( 4.5 - \frac{81}{4} \right)$ $V = 2\pi \left( 4.5 - 20.25 \right)$ $V = 2\pi (-15.75)$ $V = -31.5\pi$

Since the volume cannot be negative, we take the absolute value:

$V = 31.5\pi \, \text{units}^3$

Thus, the volume of the solid generated by rotating the region around the $y$-axis is $31.5\pi \, \text{units}^3$.

Would you like more details on any step or clarifications?

Here are 5 related questions:

1. What if the rotation was around the x-axis instead?
2. How does the method of cylindrical shells differ from the disk/washer method?
3. What is the significance of integrating with respect to $dy$ versus $dx$?
4. How would you handle different types of regions being rotated around the y-axis?
5. Can you explain the physical interpretation of the volume formula for cylindrical shells?

Tip: When integrating for volumes, always ensure that the limits and variable of integration match the axis of rotation.