To solve this problem, we are tasked with finding the volume of a solid formed by rotating the region bounded by the graph of the function $f(x) = 3x$, the x-axis, and the line $y = 6$ around the x-axis. The domain of $x$ is given as $[0, 2]$.

We will use the method of cylindrical shells to calculate the volume. The general formula for the volume of a solid formed by rotating a region around the x-axis using cylindrical shells is:

$V = 2\pi \int_{a}^{b} \left( \text{radius} \times \text{height} \right) \, dx$

Step 1: Define the radius and height of the cylindrical shells

• Radius: The distance from the x-axis to the function $f(x) = 3x$ is simply $f(x) = 3x$, which gives the radius of each cylindrical shell.
• Height: The height of each cylindrical shell is the difference between the line $y = 6$ and the graph of $f(x) = 3x$. So the height of each shell is $6 - 3x$.

Step 2: Set up the integral

Now, we set up the integral to find the volume:

$V = 2\pi \int_{0}^{2} \left( x \times (6 - 3x) \right) \, dx$

Step 3: Simplify the integrand

Simplify the expression inside the integral:

$V = 2\pi \int_{0}^{2} \left( 6x - 3x^2 \right) \, dx$

Step 4: Integrate

Now, perform the integration:

$\int 6x \, dx = 3x^2$ $\int 3x^2 \, dx = x^3$

Thus, the volume integral becomes:

$V = 2\pi \left[ 3x^2 - x^3 \right]_{0}^{2}$

Step 5: Evaluate the integral

Now, evaluate the integral at the limits $x = 2$ and $x = 0$:

$V = 2\pi \left[ (3(2)^2 - (2)^3) - (3(0)^2 - (0)^3) \right]$ $V = 2\pi \left[ (3 \times 4 - 8) - (0) \right]$ $V = 2\pi \left[ 12 - 8 \right]$ $V = 2\pi \times 4 = 8\pi$

Final Answer:

The volume of the solid is $8\pi$ cubic units.

---

Would you like a more detailed explanation of any step, or do you have further questions? Here's a list of related topics you might want to explore:

1. Understanding the method of cylindrical shells in more detail.
2. How to visualize the solid formed by rotation.
3. The disc method versus the cylindrical shell method for finding volumes.
4. Volumes of solids of revolution for different curves and limits.
5. How to calculate the volume for rotations around other axes.

Tip: When rotating around the x-axis or y-axis, always start by carefully defining the radius and height of the cylindrical shells to simplify your integration!