To find the volume $V$ of the solid obtained by rotating the region bounded by the curves $y = \frac{5}{x}$, $x = 6$, $x = 12$, and $y = 0$ about the x-axis, we can use the method of disk integration.

Step 1: Set Up the Integral

The formula for the volume $V$ when rotating a region around the x-axis is given by: $V = \pi \int_{a}^{b} \left( f(x) \right)^2 \, dx$ where $f(x)$ represents the function $y = \frac{5}{x}$, and $a$ and $b$ are the bounds of integration, which are $x = 6$ and $x = 12$, respectively.

Step 2: Substitute the Function and Bounds

$V = \pi \int_{6}^{12} \left( \frac{5}{x} \right)^2 \, dx$

Step 3: Simplify the Integral

$V = \pi \int_{6}^{12} \frac{25}{x^2} \, dx$

Step 4: Integrate

To integrate $\frac{25}{x^2}$, recall that the integral of $x^{-2}$ is $-x^{-1}$, or $-\frac{1}{x}$. Therefore: $\int \frac{25}{x^2} \, dx = -\frac{25}{x}$

Now, apply the limits from 6 to 12: $V = \pi \left[ -\frac{25}{x} \right]_{6}^{12}$

Step 5: Evaluate the Definite Integral

$V = \pi \left( -\frac{25}{12} + \frac{25}{6} \right)$

Simplifying: $V = \pi \left( \frac{25}{6} - \frac{25}{12} \right)$

Find a common denominator: $V = \pi \left( \frac{50}{12} - \frac{25}{12} \right) = \pi \cdot \frac{25}{12}$

Therefore: $V = \frac{25\pi}{12}$

Final Answer

The volume $V$ of the solid is: $V = \frac{25\pi}{12} \, \text{cubic units}$

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How would the volume change if the rotation were around the y-axis instead?
2. What would the volume be if the region were bounded by $y = \frac{10}{x}$ instead of $y = \frac{5}{x}$?
3. How does the method of shells differ from the disk method in this context?
4. What is the geometric interpretation of the volume of revolution?
5. Can you find the surface area of the solid obtained by rotating the same region?

Tip: When using integration methods for volumes of revolution, carefully choosing the method (disks, washers, or shells) that matches the problem's symmetry can simplify the computation.