To solve the problem of finding the volume of the solid generated by revolving the region bounded by the curve $y = \frac{1}{x}$, the x-axis, and the lines $x = 1$ and $x = 7$ about the x-axis, we will use the shell method.

Shell Method Overview

The shell method involves integrating with respect to $y$ when revolving around the x-axis. The general formula for the volume using the shell method is:

$V = 2\pi \int_{c}^{d} ( \text{radius} ) ( \text{height} ) \, dy$

Problem Setup

1. Radius: When revolving around the x-axis, the radius of the shell is simply $y$.
2. Height: The height of the shell at a given $y$ is determined by the difference between the x-values on the curve. Given $y = \frac{1}{x}$, solve for $x$ to get $x = \frac{1}{y}$. The height is therefore $\left(\frac{1}{y} - 1\right) - \left(\frac{1}{y} - 7\right) = \frac{1}{y} - 1 - \frac{1}{y} + 7 = 7 - 1 = 6$.

Limits of Integration

The region is bounded vertically between $y = \frac{1}{7}$ and $y = 1$. So the limits of integration are from $y = \frac{1}{7}$ to $y = 1$.

Volume Integral

$V = 2\pi \int_{\frac{1}{7}}^{1} y \cdot \left(\frac{1}{y} - 1 \right) \, dy$

Simplifying the integrand:

$V = 2\pi \int_{\frac{1}{7}}^{1} \left(1 - y\right) \, dy$

Evaluate the Integral

Let's evaluate the integral:

$V = 2\pi \left[ \int_{\frac{1}{7}}^{1} 1 \, dy - \int_{\frac{1}{7}}^{1} y \, dy \right]$

The first integral is straightforward:

$\int_{\frac{1}{7}}^{1} 1 \, dy = y \Big|_{\frac{1}{7}}^{1} = 1 - \frac{1}{7} = \frac{6}{7}$

The second integral:

$\int_{\frac{1}{7}}^{1} y \, dy = \frac{y^2}{2} \Big|_{\frac{1}{7}}^{1} = \frac{1^2}{2} - \frac{\left(\frac{1}{7}\right)^2}{2} = \frac{1}{2} - \frac{1}{98} = \frac{49 - 1}{98} = \frac{48}{98} = \frac{24}{49}$

So the volume $V$ becomes:

$V = 2\pi \left(\frac{6}{7} - \frac{24}{49}\right)$

Simplify the expression inside the parentheses:

$V = 2\pi \left(\frac{42}{49} - \frac{24}{49}\right) = 2\pi \cdot \frac{18}{49} = \frac{36\pi}{49}$

Thus, the volume of the solid is:

$V = \frac{36\pi}{49}$

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions:

1. How does the shell method differ from the disk/washer method?
2. What are some other examples of curves that can be revolved to create solids using the shell method?
3. How would the setup change if the region were revolved around the y-axis instead?
4. Can the shell method be used for regions that are not functions of x or y?
5. What are some limitations or advantages of the shell method compared to other methods?

Tip: When using the shell method, always check whether it's easier to integrate with respect to x or y, depending on the axis of revolution and the shape of the region.