The problem asks to set up and compute the integral for the volume of a solid generated by revolving the given region around the x-axis. From the image, the integral provided is:

$V = \int_0^6 2\pi y \left( 3 - \frac{y}{2} \right) \, dy$

This integral represents the volume of the solid using the method of cylindrical shells. The general formula for the volume using cylindrical shells is:

$V = \int_a^b 2\pi r \cdot h \, dy$

Here:

• $r = y$ is the radius of a typical shell.
• $h = 3 - \frac{y}{2}$ is the height of the shell.

Let's solve this step by step.

Step 1: Simplify the integrand

We expand the term inside the integral:

$2\pi y \left( 3 - \frac{y}{2} \right) = 2\pi \left( 3y - \frac{y^2}{2} \right)$

So the integral becomes:

$V = 2\pi \int_0^6 \left( 3y - \frac{y^2}{2} \right) \, dy$

Step 2: Compute the integral

We now integrate each term individually:

1. $\int 3y \, dy = \frac{3y^2}{2}$
2. $\int \frac{y^2}{2} \, dy = \frac{y^3}{6}$

Thus, the volume becomes:

$V = 2\pi \left[ \frac{3y^2}{2} - \frac{y^3}{6} \right]_0^6$

Step 3: Evaluate the limits

Now substitute the limits $y = 6$ and $y = 0$:

At $y = 6$:

$\frac{3(6)^2}{2} = \frac{3 \times 36}{2} = 54$ $\frac{(6)^3}{6} = \frac{216}{6} = 36$

At $y = 0$, both terms are 0.

So, the expression becomes:

$V = 2\pi \left( 54 - 36 \right) = 2\pi \times 18 = 36\pi$

Final Answer:

The volume of the given solid is $\boxed{36\pi}$ cubic units.

Would you like more details or have any questions?

Here are some related questions:

1. How do you derive the cylindrical shells method formula?
2. What other methods can be used to find the volume of a solid of revolution?
3. Can this problem be solved using the disk method instead of shells?
4. How do the limits of integration change if the region is revolved around the y-axis?
5. What if the function being revolved were more complex—how would that affect the setup of the integral?

Tip: Always carefully check the radius and height in cylindrical shells problems, as they vary depending on the axis of revolution.