To find the volume $V$ of the solid obtained by rotating the region bounded by the curves about the $x$-axis, we can apply the Disk Method.

Step-by-Step Approach:

1. Identify the curves and region:

   • The region is bounded by:
     • $y = x - 1$
     • $y = 0$ (the x-axis)
     • $x = 8$

2. Set up the Disk Method formula:

   The volume of a solid obtained by rotating a region about the $x$-axis is given by: $V = \pi \int_{a}^{b} [f(x)]^2 \, dx$ where $f(x)$ is the function describing the curve being rotated, and $[a, b]$ are the limits of integration along the $x$-axis.

3. Set up the integral:

   The curve $y = x - 1$ will be rotated about the $x$-axis. This gives the radius of each disk as $f(x) = x - 1$. The limits of integration are from $x = 1$ (where $y = 0$) to $x = 8$ (as given in the problem).

   So, the integral becomes: $V = \pi \int_{1}^{8} (x - 1)^2 \, dx$

4. Expand the integrand:

   First, expand $(x - 1)^2$: $(x - 1)^2 = x^2 - 2x + 1$ Now, the integral becomes: $V = \pi \int_{1}^{8} (x^2 - 2x + 1) \, dx$

5. Integrate:

   Now, integrate each term: $V = \pi \left[ \frac{x^3}{3} - x^2 + x \right]_{1}^{8}$

6. Evaluate the definite integral:

   First, evaluate at $x = 8$: $\left( \frac{8^3}{3} - 8^2 + 8 \right) = \left( \frac{512}{3} - 64 + 8 \right) = \frac{512}{3} - 56 = \frac{512 - 168}{3} = \frac{344}{3}$

   Next, evaluate at $x = 1$: $\left( \frac{1^3}{3} - 1^2 + 1 \right) = \left( \frac{1}{3} - 1 + 1 \right) = \frac{1}{3}$

   Now, subtract the two values: $V = \pi \left( \frac{344}{3} - \frac{1}{3} \right) = \pi \left( \frac{343}{3} \right)$

7. Final answer:

   The volume of the solid is: $V = \frac{343\pi}{3}$

Do you need further details or clarifications?

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Here are 5 related questions to expand on the topic:

1. How does the Disk Method differ from the Shell Method?
2. What would the volume be if the region were rotated about the line $y = 1$?
3. Can you solve the same problem using cylindrical shells instead of disks?
4. How would the volume change if the curve were rotated about the $y$-axis?
5. How do you find the volume of a solid with a more complex boundary, such as $y = x^2$?

Tip: Always check the limits of integration carefully when setting up an integral for volumes of rotation.